A B737 CREW; AFTER LNDG AT DEN; CLRING THE ACTIVE; INCURRED THE PARALLEL TXWY BEFORE CONTACTING GND CTL.

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AFTER LNDG ON RWY 17R DURING A SIGNIFICANT WX EVENT JUST S OF DEN; TWR CTLR ADVISED TO EXPEDITE TO FIRST FORWARD HIGH SPD TXWY DUE TO OTHER ARRS AND DEPS DIRECTLY BEHIND US. WHILE TRYING TO COMPLY WITH TWR CTLR'S INSTRUCTIONS WE EXITED AT M5 HIGH SPD TXWY AND ATTEMPTED TO TAXI AS FAR FORWARD AS POSSIBLE IN CASE OTHER ARRS NEEDED THE SAME TURNOFF; SO AS TO NOT PREVENT OTHER ACFT FROM CLRING THE RWY. SEVERAL MULTIPLE CALLS WERE MADE TO GND CTL WITH NO ANSWER AND CONSTANT COM WITH ACFT TAXIING FOR DEP. DUE TO THE ANGLE OF THE HIGH SPD TXWY; THE FO WAS NOT ABLE TO SEE AN ACFT THAT WAS TAXIING SBOUND ON TXWY M. WHEN THE TURN WAS STARTED TOWARD TXWY M AT THE END OF THE M5 HIGH SPD; THE ACFT WAS FIRST SPOTTED ON TXWY M SBOUND. DUE TO OUR TAXI SPD; COMPLYING WITH TWR INSTRUCTIONS; WE WERE UNABLE TO STOP BEFORE STARTING TO ENTER TXWY M. BOTH US AND THE SBOUND ACFT STOPPED. SEVERAL MINS WERE SPENT TRYING TO TALK TO GND CTL WHEN THE SBOUND ACFT WAS ABLE TO GET THE ATTN OF GND CTL; THE CTLR ACTED SURPRISED THAT BOTH ACFT WERE WHERE THEY WERE AND THAT A POTENTIAL CONFLICT HAD OCCURRED. IT WAS VERY OBVIOUS THAT BOTH THE TWR AND GND CTLRS WERE COMPLETELY OVERLOADED. IN THE FUTURE; IT WOULD BE VERY HELPFUL IF GND CTL AT DEN COULD SPLIT INTO ARR AND DEP FREQS DURING VERY BUSY OPS DURING WX EVENTS. ALSO; TWR CTLRS NEED TO KNOW WHAT POTENTIAL PROBS MAY EXIST WHERE AN ACFT HAS BEEN REQUESTED TO EXIT THE ACTIVE RWY.