Should be an interesting discussion.... proceed
I'll start (although my kids are older (32 & 22) so my mistakes are baked in)...
We didn't raise them in a church although we attended a Unitarian Universalist church for a little while.
We went to museums to instill a fascination with science and the natural world.
We went on hikes in the woods and collected fossils when the opportunity arose.
We had family dinners every night and asked (and still ask) "What was your most exciting thing today?"
They've grown up to be decent human beings with empathy and be able to take joy in life.
I think we did okay.
PS. That all may not sound "Epicurean" and, yes, I didn't discover Epicurus until well after our children were raised. Does that mean I had an "Epicurean" approach, unbeknownst to me? What does it mean to parent one's children "Epicureanly"? That's what I hope we explore in this thread.
I just want to preface this thread by saying that I think there are many really great ways to parent children, but it can be a touchy subject. I hope that we can all respect and learn from each other.
I'm sure many of us are all of one mind that we shouldn't present myths as facts about reality, but I think tying mythos to imaginative play is one way to introduce mythical concepts to children. Imagination play is important at the critical young ages of about 2 to 7. (Here's a Scientific American Article on this.) The inevitability that my kids will at the very least be exposed to religious stories and ideas means I try to be proactive in introducing them to these concepts. Both of my kids are under 8 years old and I use folklore creatures in imaginative play to explore "magic" and "mystery" that they are already exposed to from the broader culture. I bring the kids half-way in on the subterfuge by having them take part in making the magical games playout for their sibiling for example. A gnome, brownie or mermaid I feel is more approachable and not so mentally overpowering as Gods; because modern lore about them and the lore we homebrew, doesn't convey "power over" dynamics, but are more mischieveous or have "power with" dynamics that models the sorts of social power I'd like my kids to exercise. (referencing Mary Parker Follet's work here) I see them both often grappling and playing with notions of "infinity" and huge numbers, and attaching an all-knowing, all powerful personality that could be angry with them to that sort of thought seems a bit cruel to me. Another upside to Epicurean theology here. Folklore creatures also tend to be more natural-world oriented which has drawn my kids' attention away from the magical worlds in media like video games and movies towards creating a little magic for them out in the natural world. So in this way I feel like I am teaching them that the magic that exists is what we create for ourselves and other people.
As for the afterlife, that one is a bit trickier as my oldest has a bit of death anxiety. So even though I generally tell them many things leading to the idea that we cease to be after death and the benefits of believing that, they do explore afterlife narratives. I think it's fair for a young one to entertain these idealistic notions in coming to terms with death. Dreams are often a big part of discourse with the kids so I will talk about how the deceased, as well as mythical creatures, can appear in our dreams, but I don't tie these things to anything supernatural. These things are part of the human experience, so I should talk about them as possiblities. Also, that we all change over time and our former selves "die" as we take on new roles or resonsibilities in this life; hinting that this may explain some concepts about the afterlife. I suppose in these ways I have the kids entertain the possibility of idealistic myths, but also naturalize these ideas.
That all sounds great! I am doing UU with my two as well right now, and we also try to do nature hikes and museums trips. Being intentional about meal time seems very Epicurean to me. I try to do something similar though we don't always have the time, schedule or energy to make it happen every night. We do try to make it special when we do eat at the table with less direct, dimmer lighting and have a candle going with some music. I'm reading a book about the Danish notion of 'hyyge' (HOO-gah) which (as I understand it) describes the atmosphere of coziness and coviviality with good company that also encompasses ideas about lighting in a living or social space, and I am having fun trying to get the lighting in my house just right for meal time. Maybe another thread exploring that sometime...
Anyway, this is getting long. I'll leave it at that for some more back and forth before introducing any new ideas. Feel free to jump right in to more decidedly Epicurean parenting discussion. I will be gathering my thoughts on that for another post soon.
Thoroughly enjoyed your post, Root304 .
As a point of departure for further posts on this thread, I tried to find specific references to children or child-rearing in the extant texts:
Epicurus's will and other citations in Diogenes having to do with child rearing and their education:Quote from Diogenes Laertius 10.19-23, 85
"And let Amynomachus and Timocrates take care of Epicurus, the son of Metrodorus, and of the son of Polyaenus, so long as they study and live with Hermarchus. Let them likewise provide for the maintenance of Metrodorus's daughter, so long as she is well-ordered and obedient to Hermarchus; and, when she comes of age, give her in marriage to a husband selected by Hermarchus from among the members of the School ; and out of the revenues accruing to me let Amynomachus and Timocrates in consultation with Hermarchus give to them as much as they think proper for their maintenance year by year.
 "Let them make Hermarchus trustee of the funds along with themselves, in order that everything may be done in concert with him, who has grown old with me in philosophy and is left at the head of the School. And when the girl comes of age, let Amynomachus and Timocrates pay her dowry, taking from the property as much as circumstances allow, subject to the approval of Hermarchus...
 And when near his end he wrote the following letter to Idomeneus :
"On this blissful day, which is also the last of my life, I write this to you. .... But I would have you, as becomes your life-long attitude to me and to philosophy, watch over the children of Metrodorus."
Many others besides you will find these reasonings useful, and especially those who have but recently made acquaintance with the true story of nature and those who are attached to pursuits which go deeper than any part of ordinary education (ἐγκυκλίων). So you will do well to take and learn them and get them up quickly along with the short epitome in my letter to Herodotus.
62. If parents have cause to be angry with their children, of course it is foolish to resist, and thus not try to beg for forgiveness. But if they do not have cause and are angry without reason, it is ridiculous to make an appeal to one who is irrationally opposed to hearing such an appeal, and thus not try to convince him by other means in a spirit of good will.
εἰ γὰρ κατὰ τὸ δέον ὀργαὶ γίνονται τοῖς γεννήσασι πρὸς τὰ ἔκγονα, μάταιον δήπουθέν ἐστι τὸ ἀντιτείνειν καὶ μὴ παραιτεῖσθαι συγγνώμης τυχεῖν, εἰ δὲ μὴ κατὰ τὸ δέον, ἀλλὰ ἀλογώτερον, γελοῖον πᾶν τὸ πρὸς ἔκκλησιν <ἐκκαλεῖν> τὴν ἀλογίαν θυμῷ κατέχοντα, καὶ μὴ ζητεῖν μεταθεῖναι κατʼ ἄλλους τρόπους εὐγνωμονοῦντα.
From Epicurus Wiki
While Epicurus was evidently a devoted and thoughtful son to his mother, and charitable towards his friends' children, the paucity of sources does not allow for a fully developed understanding of his views on parenthood in general. Some minimal, practical advice is offered in Vatican Saying 62, although it is always questionable whether this text (like others in this collection) is an authentic, first-hand expression of Epicurus' own views on the matter.
As with marriage, Epicurus approved of parenthood, in certain circumstances.
Reflecting the standard mores of Roman society, Lucretius unfolds more elaborate concerns about children, parents, and the relationship between the two in De Rerum Natura; he also reveals a warmer attitude towards parenthood in the broadest sense, as he waxes poetic on parental relationships throughout the animal world.
. . . . . .
Were known; and when they saw an offspring born
From out themselves, then first the human race
Began to soften. For 'twas now that fire
Rendered their shivering frames less staunch to bear,
Under the canopy of the sky, the cold;
And Love reduced their shaggy hardiness;
And children, with the prattle and the kiss,
Soon broke the parents' haughty temper down.
Then, too, did neighbours 'gin to league as friends,
Eager to wrong no more or suffer wrong,
And urged for children and the womankind
Mercy, of fathers, whilst with cries and gestures
They stammered hints how meet it was that all
Should have compassion on the weak.