Is it legal for someone to post pictures of minors without parental consent on Facebook on a private Youth Christian FB page?

Having such permission would be a best practice, yes.

It sure wouldn't hurt! You never know what an irked parent will do. You should get a model release for each kid, signed by the parent.

It would be to have a written policy on what gets posted, how, when, etc. for the group to follow. In addition, when you're signing up kids for the youth group, have a form that explains your policy on posting photos, etc. that parents can read, understand, and sign to allow or disallow their kids' images/names to be posted.
Contact other churches and ask if they have a policy. If so, explain what you're doing and ask if you could get a copy of their policy and forms to use as a template.

You do not need permission to post otherwise lawful pictures of minors on FB. If that were required, almost every FB account holder would be running afoul of the law.
I think you should stand ready to remove them should a parent complain, but so long as they are your pictures they do not have a legal cause of action merely because you posted them.
As my colleagues note, getting permission certainly can't hurt and it would help reduce any complaints upfront.
Best regards,
Frank
Natoli-Lapin, LLC

I would send a strongly worded letter requesting the pictures be removed immediately.

If any of the teenagers pictured are under the age of 13 then YES you do need the parent's consent.
Those pictures would fall under the COPPA guidlines and you should have a lawyer help you with devising the type of consent you need.