As I understand it, the guns made in grades with twist barrels were not held for a long time in inventory. So I expect the gun was actually made within a year (or less) of the date indicated by the serial number.
A numeral 5 (with super script 8) usually tells us the weight of the barrels before they were struck. That sounds light for a 10 or 8 gauge.
That numeral 3 at that location suggests this will be a size 3 frame.
At the start of Parker Brothers history, grades are not stamped on the barrel flats nor on the water table.
There may be factory records. You can find out about records availability when you go to the page on this website that talks about research letters. There is a search function for you to input your specific serial number to learn whether specific records are available.
Your gun’s serial number may also be in the Parker Gun serialization book, by Price & Fjestad. You can share the serial number here for that information (assuming it is in the book). The book lists a gauge, information about the stock and extras, as well as the original barrel length.
https://www.amazon.com/Parker-Identi.../dp/1886768374
If you put some pictures up, a lot more commentary as possible.