I'm most definitely in the camp of children should send out thank you letters after Birthday's & Christmas.
BUT
Should a child send a thank you letter to a parent, especially a non resident parent? Although resident or non resident really shouldn't make any difference.
I was never expected to send thank you letters to my parents as a kid, although sometimes when they bought something really fabulous I would send a gushing thank you to them, but it was never expected by them. I've never expected letters from my kids.
My ex has got a real bee in his bonnet about never getting them. He hardly ever sees the kids, I'm talking like 3 or 4 visits a year for a few hours and never buys them presents they like due to not having a clue what they are into, although I appreciate the fact that a child likes the present or not shouldn't influence whether they send a thank you or not.
So in essense, should children send either parent a thank you letter?