The '65's are notorious for having lousy ground connections in the sockets to the socket housing in the trunk lid and quarter panel extension. They have to be fully seated in order for the light to have a good ground and function. A $10 circuit test light is your best friend when working with 50 year old cars and their wiring. Use it find out where the bad connection is causing the right side tail lights not to work. ( A hint here. If one tail light is working, your fuse is not burnt. Look elsewhere. )
Turn on the lights, use the test light to see if you power to the light sockets. The right side have a green wire and a brown wire. Can't remember which one is brake/turn signal and which one is tail lamp. But, the test light should light up. If not start working back towards the tape wire harness that goes under the seats and carpet. Most likely your bad connection is before that.
I finally gave up on the '65 era light sockets and got a bunch for a '73 Monte Carlo because that era use a ground wire instead trying to ground the socket itself. No problems since. Also, make sure you have the dual element # 1157 bulbs.
1965 Impala SS, 383, 4 spd
1970 Caprice LS-4 454, TH400