If the reverse lights are on the LOM (lamp out module) sensing circuit, then you don't want a second parallel current draw, should only use the reverse wires (in addition to their existing purpose) to trigger a relay to turn on the led flush mounts from an alternate, fused power wire you add.
OR... you could just try it and see, except that even if it works, you should disconnect the reverse lights to see if the LOM detects they aren't working instead of being fooled by the alternate draw from the flush mounts. This assumes you have added up all the current the circuit is capable of and there is enough margin for the additional amount the flush mounts draw. I don't know the existing current draw nor that for the flush mounts. You might need to look at the fuse rating and wire gauge as well.