I used to do a lot of Parkerizing.

The problem you are going to run into is time after parking. You have to shoot your Moly IMMEDIATELY after parking the parts. Like park them, rinse them, bake them and shoot them.
That is the only way you can be assured that absolutely NO oil has gotten on them.

If you send the parts out to be parked they will likely be oiled, otherwise they will rust. Yes, parkerized components will rust if not oiled. Even if they aren’t oiled they will wick oil from anything they contact, they would have to be kept surgically clean.
Phosphate coating either zinc or manganese is essentially a sponge for oil. I will typically soak parts for 2-3 days in used motor oil and 2 days after pulling them out they look dry.
If the parts get ANY oil on them before coating them you will NEVER get it all out. You can bake it, soak it in acetone and it will still leach oil.

The park is a great base for an applied finish but it is not mandatory.
I used to shoot lots of Moly Resin without parking and never had any issue with longevity or rust/corrosion.

Prep is the key, disassemble completely. Degrease thoroughly. When you think it is clean do it again.
Bake the parts at 350 for an hour or two. Let it cool, degrease again.
Media blast, 80-100 grit aluminum oxide or similar works well- don’t beadblast it leaves a finish without any “tooth” to hold the coating.
Degrease it and bake it again.
Re blast if any hints of oil show, it will appear as dark spots.
Degrease again (seeing a trend?)
Bring the parts to about 150 degrees and then shoot your Moly.
Let things dry for about an hour and the do the bake to set the Moly.
Let it cool off overnight at least before reassembling, give it a chance to set and cure completely.
FWIW the prep for just parkerizing is just about the same, one bit of oil will kill the finish and your park solution.

It is time consuming and a PITA to do correctly.

If this is a one or two time thing I would outsource the entire job.