What's the best way to record payments in installments?
I would consider using Membership Types rather than Pledges. You could set up each payment term as its own Membership Type, with different terms (ie 1-month membership that lasts for 10 months; 6-month membership; etc). Then create Membership Contribution pages for each term. You could turn on the auto-renew feature, or make that optional and setup payment reminder emails. 

You could also create a new Financial Type called Registration Fee (or something similar) so that it's clearer what the payment is for. Even if you already have memberships set up for something else, you can set the system to allow multiple memberships, and just change the wording on the public-facing pages so it doesn't confuse the parents.

Caveat - I haven't set up CiviCRM this way before, but am currently considering it for one of my clients that is a preschool. I played around in the demo a bit to see if this would work as I'd like it to, and so far haven't run into anything that would make it seem otherwise.