Don, going from memory (too tired to look it up again), I think that:
(1) yes, you can send First Class Parcels that are under 3/4" thick, no problem.
(2) But if you have a Delivery Confirmation (Tracking) label on the First Class parcel mail-piece, and it is under 3/4", the Post Office will theoretically charge the recipient a Priority Mail rate (or maybe return it).

At least those are the old rules.

Robert