Would that work and get around having parents see my personal page?
Yep - you can also go one further :-) If you search for your day care parents FB account, you can actually block them. You can also block them using their e-mail addresses if they use the same e-mail address to log into their FB account - that's a bit hit and miss though.

Blocking anyone will hide you completely from them. Even if you have a mutual friend and both comment on their page, they will not see the comment you made.

You won't come up in any search they perform and any comments you make in public groups cannot be seen by those you have blocked.