What kind of solution should I use to clean with.
Looking for something that doesn’t effect bluing, stainless steel
or parkerizing..

I play it safe and use Hornady's. It seems expensive at first, but you can reuse it. I drain and strain the mix into a 2 1/2-gallon bucket and put the lid on between uses. It does a good job. For drying I use an old food dehydrator for both gun parts and shell cases. I also use Hornady's case cleaner. After drying I use a CLP product to make sure everything has a coating/drive out any hidden moisture, then wipe and use gun lube or grease on the appropriate points for the specific weapon.

Just noticed you are in CT. I'm in SE CT, a Navy transplant originally from KY.

Simple green has worked well for me.   Aluminum “safe” is what I keep around but regular will also work for steels.

Other then buying a commercial solvent,............

Dawn and hot water.
Cheap paint thinner....Careful about sparks.
Liquid kitchen cleaners.
CAREFUL with harsh soap cleaners like standard Simply Green, Purple Power, Greased Lightning etc...Most WILL attack and damage aluminum and anodized finishes and do so faster in a US cleaner.