I always assume that gifts are meant to be "from" the children, so I thank the children there and in person. I wouldn't then thank the parents, as that's then sort of acknowledging that it's not really from the child in the same way. I know it's a pretence that the child arranged it, but it's part of gift giving. I wouldn't always then thank the mum - which assumes that it was her job rather than the dad's - though if the parent collected the child and I had a chance, I might thank them as well. Or if I had something by email from a parent's account, I'd write back to them to say thanks. But otherwise, thanking the child is perfectly normal.
I wouldn't change my thanks depending how expensive a present it was. I wouldn't even know a Yankee candle was expensive. I thought it was a cheaper version of a fancy candle, but I really don't know.
I would think that if they are thanking me for teaching or coaching their child, and I then thank the child for a gift, that is enough back-and-forth. Otherwise, they might then have to reply to my thank you card/text and then I'd feel obliged to reply to their reply, which would undoubtedly wish me good holidays or ask how things were, and then I'd wish them that in return, and then they'd probably acknowledge that, and so on!!
Much simpler to thank the child in person, and to assume that the parents wanted to thank me in the first place and weren't doing it for acknowledgement. I do appreciate them hugely, so I would not assume that a lack of an additional thanks means that they do not appreciate them.
I'd never put them on social media, and especially not where others could see; that does seem wrong to me. A gift to the club as a whole or something, maybe, but not in the way you've described.