I respectfully disagree...

It is much too rare a Parker to do as suggested above.

That is an exceptionally beautiful gun, and as Dean says, quite rare. The 3 on the bottom of the barrel lug is the frame size, and the 5 on the barrel flats is the barrel weight. It is an A grade, and if I'm not mistaken, post dates the time when they were graded by their cost.
Please don't be tempted into even the slightest bit of cleaning. When you seek out the advice of experts, you will be hit by sticker shock at what a proper restoration will cost, but believe me, it is a fraction of the guns potential value. A good cleaning by an experienced expert may be all it needs. If the lockplate engraving is in the same condition as the receiver, you don't need anyone chasing the engraving. Thank you for sharing it with us.

I believe the unstruck barrel weight is 5# 6 oz.. The frame size is indeed #3.
I think the 5 on the barrel flats is the equivalent of a B grade lifter but I am no expert on high grade Parkers. All in all a beautiful Parker.
Best regards,
Erick

I stand corrected. SN 20424 is listed as a D6.

The DD on the barrel flats indicates Finest Damascus and I believe the 5 indicates 5-Iron Crolle Damascus.

That is indeed a very beautiful gun! It has been stored incorrectly for some period of time, but is absolutely an honest gun. Just remarkably un-messed with.
This is one of those times that you just take a deep breath, and proceed slowly, and with caution. I'm not qualified to tell you what those steps are, but there are those here that are.
You certainly made the right first step by coming here.

In the book as a grade 6.

Wow! there seems to be a circle behind the hammer what's in it. More close ups of lock plates and 4end wood please. What a gun. ch

Thanks so much for the input. I will begin to do my homework!
My friend was left several other guns by her late father. Her siblings want them sold. The list includes: two Winchester model 1895's (one has saddle ring missing)
Win model 70 .30-06 ser #258184 (1953)
Rem model 8 .35 Rem. Ser#58566
AH Fox s/s. 12Ga
Rem 742 .308
Rem 7400 carbine .30-06
FN Browning Auto 5 Sweet 16
Rem hump back 12 Ga
Savage .35 Rem pump
Colt SA Army .38Spl Ser#25512SA (1959) (slight 1/2" holster wear both sides of muzzle)
S&W DA .45 with lanyard ring Ser#3882054 (Perfect)
Colt Gov. US Marine Corps Iwo Jima Commererative in wood/glass case with all papers and white gloves for handling (Perfect. Never fired)
48 Star American Flag
Japanese Flag bravely taken by her Dad in combat (USMC) on Iwo Jima (the person he got it from didn't need it anymore) Perfect condition.
Collection is in Massachusetts.
She is still finding guns in his house and barn as I write this.
Thanks for your help.

Sounds like the Mother Lode!

When she opened the closet doors in her Dad's house, I felt like a kid in a candy store!
I was given my first shotgun ( single shot .410) by my Dad when I was 10. He raised bird dogs, and I could never replace the memories we shared in the field until he passed away at the age of 53. Most people in Massachusetts could never understand a redneck like me. I have been collecting guns ever since, but I have never seen workmanship like this Parker.