This is the chair.
My parents gave it to me.
This is the chair which my parents gave to me.
This is the chair which my parents gave me.
Should it have "to" or not?
I think the better question would be whether or not you should have the "which". Personally, I'd leave it out:
This is the chair my parents gave me.
This is the chair my parents gave to me.
Getting back to your original question, I prefer the shorter version of those two (I'd recommend leaving out the "to," too). However, the second sentence doesn't sound particularly jarring or ungrammatical, and would be acceptable in conversation.
Perhaps the most direct way of all to say it:
My parents gave me this chair.
but I realize there are some contexts where the most direct way isn't always the very best way to word something.