Do any of you give probiotics to your breeding tiels and to their babies?

No, it isn't necessary unless they are on antibiotics.

I don't unless there's a reason for it. Last year one nestbox started to smell bad which can be a sign of yeast or other problems, so I applied some home remedies including probiotics and it cleared up.

It wouldn't hurt to provide a little if you want to. Breeding is stressful on the parents and this can make problems flare up. I think that's what happened with my pair last year.