Instrument bulb LED replacement
That will be fine.

A minor correction on the calculations: since the LED will drop 2 V that only leaves 12 V across the resistor. So \$ R = \frac {V}{I} = \frac {12}{0.06} = 200 \; \Omega \$. 220 Ω will be fine.


  Would putting this LED / resistor in place of an incandescent bulb affect the other bulbs on the circuit?


There are two circumstances where this might matter:


Series connected lamps. This doesn't apply in your case.
Indicator (turn-signal) lamps. Older cars used a bi-metallic strip contact for the flasher. The current ran through the contact and when it warmed up (in about half a second) the contact opened turning the lamp off. It cooled down in about the same time and turned back on again. They had the nice feature that if one bulb failed the flash rate would slow down to about half and alert the driver to the fault. Some electronic flashers mimic this behaviour. Given that you've chosen red LEDs it is unlikely to be a turn signal indicator so it shouldn't be a problem.


In general, parallel lamps don't affect each other. (All your house lamps of different power ratings are wired in parallel.)