Corporate jet Captain reported descending below cleared altitude approaching MMTO in a high terrain environment after misunderstanding an ATC clearance.

Narrative

This was our third leg of the day in Mexico that had numerous immigration issues at all stops. The first leg was a deviation due to thunderstorms over the field and Mexico Approach Control training. We were cleared the Lenul 2A arrival into MMTO. My copilot was on a different frequency verifying we did not have to clear customs now that we were in range and could be heard. Between TO603 and TO604 we were at 15000 [ft.] and the Controller stated 'Cleared direct GAINY descend to 13;000.' We entered direct GAINY and started the turn and also initiated a descent to 13000. AT 14;200 the Controller asked if we were VMC. We replied no and he told us to climb immediately to 15;000 due to terrain. He then mentioned that we were not cleared to 13;000 until after GAINY. We climbed to 15;000 and maintained that altitude until he cleared back down to 13;000. Nothing else was mentioned and we were handed off to Tower. Be terrain aware and always double check altitudes if the clearance doesn't make sense. Mexico Controllers phrase clearances and directions slightly different. Take the time to verify instructions.