I concur 100% with the previous posts re: strip and cleaning a Parker Hammerless double, whether it is an ejector or extractor gun. And I favor OOOO grade steel wool and both Kroil thinned slightly with acetone or WD-40, plus Hoppe's No. 9 and Rem Oil aerosol.
Old Boy Scout survival tip- carry several wads of OOOO steel wool in your emergency firestarting kit- it has just enough oil in its composition to aid in starting a fire, that same oil will also assist in removing minor crud and rust. An older typewriter eraser- hard to find in today's non-typewriter using era, soaked in Hoppe's is also a mild abrasive and can work.
The heads of all the machine screws seen on your Parker are one of the many things Parker experts examine when appraising your gun. Indexing or clocking of the screw head slots, meaning they all faced in the same direction (like 12 to 6 o'clcok) and no mars or boogered slots.
Those two "no-no's" are a sure sign of a ham fisted amateur trying to field strip a Parker (or other fine doublegun). I would wait for the posting on the FAQ from John Dunkle, and also suggest you spend a few $ on at least two things: a duplicate set of Grace hardened and ground gunsmith screwdrivers (you won't fine any Phillips head tips in their mix- Brownell's is a good source, also Midway)-- reason for the second set- because that way you have enough to allow you to grind the shank and tip, if needed to an exact fit for special screws you may encounter. Most have a tapered slot, and the closer to perfect fit you have with your "turnscrew" the less chance you have of a slip and mar result- NOT GOOD.
Second is a set of small magnets- as all Parker metal parts (except perhaps the sight beads) are from forged steel, magnets are perfect for your gun bench to keep small machine screws, pins, springs etc. from getting lost.
A well padded vise, a gunsmith soft tip or brass hammer, various drift and pin punches- and a lot of patience are also important.
If you have a competent gunsmith in your area that is quaified to work on Parker guns, perhaps you could offer to pay him for a "hands on dis-assembly/re-assembly" drill- not a complete tear down to all parts, but enough so you can clean and inspect your gun with a solid degree of confidence.