Thanks rosewood for the photos of your inherited Nylon 66. Indeed that is chrome so it is most likely a Apache Black and Chrome (sales code N66AB) of which some 221000 were made from 1962 to 1984. From your photos it looks like your manufacture code is P-P which indicates June 1967. I agree that it appears your barrel s/n does not exist which while not earth shaking, is just before the October 1967 start of 400000 bbl numbering series. Probably the chrome was harder to roll stamp than the blued carbon steel but that is just a guesstimate on my part.
Can you confirm the date code on yours as P P? The 40 is just the inspectors number. The photo sure looked like P P to me but you may be able to see something different up close.
Does June of 1967 (or sometime thereafter of course) sound like a time frame that your dad would have bought the rifle?
Another thing I noticed was that your barrel logo has the stylized "Remington" between "pat pend" and "22 LR ONLY"
which places yours after 1964 as the rifles before that did not have the Remington like my 1960 Seneca Green.
In looking into the Apache Black and Chrome I did see a example with date code AM (March 1965) also with no barrel serial number, and another dated 1973 with a clear serial number (2298957) on the left side of the chromed receiver cover so it looks like you have a nice early style rifle.
Enjoy that rifle...they are fun to shoot (and collect..still inexpensive as firearms go)