Jeff, That is a nice back action Parker your Grandfather left you. I would put the manufacture date at 1872 based on what I have read. Overall the condition is pretty good. Most back actions I see are very rough.
I have a few that I shoot with black powder and smokeless loads. I have Serial#s 01079 and 01081, not too far from yours. Does a 12 gauge shell fit? Some of these early guns are in a few other gauges: 10, 11, 12, 12b, 14 so that would be nice to confirm.
Then I would gently clean it and go use it as Parker hammer guns are just too much fun to shoot
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