Pull-up resistor power rating on signal wire in automotive?
Nasty voltage spikes on a car "12 V" system are real, but their average power is small.

During normal operation, your resistor could dissipate (13.6 V)2/(1 k&Omega;) = 185 mW.  In theory, the next common size of &frac14;W would cover it.  However, cars get hot.  Take a look at the datasheet of a &frac14;W resistor to see how the power dissipation capability is derated with temperature.

Short spikes on the "12 V" line don't add much average power, but do represent some fixed energy per spike that the resistor has to absorb effectively instantaneously.  The power dissipation capability of the resistor is mostly irrelevant.  It's the thermal mass that decides how much the resistor will heat for a given energy spike.

Cars are a harsh environment.  A &frac14;W resistor doesn't leave much margin, even at a convenient ambient temperature.  I'd go with the next common larger size, which is &frac12;W.