WHILE ATTEMPTING TO PASS A PARKED ACFT ON A CROWDED RAMP THE CAPT OF A B757 BRIEFLY EXCEEDS THE PAVED SURFACE WITH THE LEFTMOST MAIN GEAR TRUCKS.

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WE LANDED WITHOUT INCIDENT AT MGGT AND TURNED OFF AT TXWY U. IN ORDER TO PROPERLY CLR THE RWY; ACFT NEEDED TO BE TAXIED BEYOND THE PARALLEL TXWY ONTO THE RAMP ENTRANCE. GND CTL ASSIGNED US GATE X ON THE OTHER SIDE OF THE TERMINAL. ALTHOUGH THERE WAS A B737 AT GATE Y; I FELT THERE WAS AMPLE ROOM TO TAXI BEHIND IT TO THE S APRON. I BEGAN MAKING A L TURN AND GND CTL ASKED IF I HAD ENOUGH ROOM. I SAID YES AND 2 WING WALKERS SOON APPEARED TO GUIDE MY R WING. BEING OVERLY CAUTIOUS; I CUT THE CORNER A LITTLE TOO MUCH AND THE OUTBOARD TIRES OF MY L LNDG GEAR BRIEFLY CUT THROUGH A GRASS AREA. A GEAR INSPECTION AFTER PARKING INDICATED NO DAMAGE. IN RETROSPECT; A CLRLY BETTER CHOICE OF TAXI RTE WOULD HAVE BEEN TO ENTER THE NORTH RAMP; DO A 180 DEG TURN AND EXIT THE RAMP VIA TXWY U THEN BACKTRACK TO TXWY T VIA THE PARALLEL TXWY OR THE RWY ITSELF. CONTRIBUTING TO THIS EVENT WAS THE FACT THAT BOTH PLTS WERE RELATIVELY UNFAMILIAR WITH THE ARPT. THIS WAS MY SECOND TRIP INTO MGGT; AND IT WAS THE FO'S FIRST TRIP IN A B757. ADDITIONALLY; I FEEL THAT THE CUMULATIVE EFFECT OF FATIGUE AND STRESS WERE A FACTOR CONTRIBUTING TO IMPAIRED JUDGEMENT FOR BOTH MYSELF AND THE FO AS BOTH OF US HAD BEEN SUBJECTED TO THE EFFECTS (PREPARATION AND AFTERMATH) OF TWO SOUTH FLORIDA HURRICANES WITHIN 2 WKS OF EACH OTHER WITH ANOTHER HURRICANE POSSIBLY HITTING SOUTH FLORIDA WITHIN A FEW DAYS. BOTH OF US LIVE IN SOUTH FLORIDA.