I have the opposite issue with my car (70 Challenger). The reverse light stays on in every gear except park. Off in park but on in R, N, 1,2,3 with RMVB and a transbrake.

1963 Belvedere 440 Max Wedge Tribute
1970 Charger R/T S.E. 440 Six Pack
1970 Challenger R/T, 528 Hemi
1970 Charger 500 S.E. 440 4 BBL
1970 Plymouth Road Runner 383
1974 Chrysler New Yorker 440
1996 2500 RAM 488 V-10 4X4
2004 3500 Dually Cummins 4x4
2012 Challenger R/T Classic.