Parallel RL Time Constant
Putting another inductor (of any inductance) in parallel with the existing one will reduce the total inductance, so the L/R time constant will be reduced  if R stays the same.   

If the objective is to get a specified current through a specified inductance and resistance within a short time, increasing the voltage will reduce the time for the current to rise from (say) zero to a given current. It does not change the time constant (which is the time for the current to rise to about 63% of the final current), but it makes the current rise faster.