DURING A PLT TRAINING FLT IN A C172; THE INSTRUCTOR PLT BELIEVED THAT HE WAS CLRED ACROSS AN ACTIVE PARALLEL RWY TO PARKING IN FRONT OF ANOTHER ACFT CLRED TO LAND ON THAT RWY. HOWEVER; HE DID NOT HEAR THE TWR'S INSTRUCTIONS TO HOLD SHORT DUE TO THE XMISSION BEING 'STEPPED ON' BY ANOTHER ACFT.

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AFTER LNDG ON RWY 1L; WE PULLED OFF ON A TXWY BTWN THE RWYS (1L AND 2R) AND WERE TOLD TO CONTACT CONCORD GND. (RWY 1R WAS TECHNICALLY CLOSED AS SUNSET HAD JUST OCCURRED AND RWY 1R HAS NO RWY LIGHTS). HOWEVER; AN ACFT HAD ASKED TO LAND ON RWY 1R AND WAS TOLD HE COULD DO SO AT HIS OWN RISK. I OBSERVED THIS ACFT ON ABOUT A 1 MI FINAL AND KNEW WE HAD PLENTY OF TIME TO TAXI ACROSS RWY 1R WITHOUT DANGER. WE CALLED CONCORD GND; STATING OUR LOCATION BTWN THE RWYS AND ASKED FOR TAXI CLRNC TO THE E RAMP. WHEN THE GND CTLR ANSWERED US; SHE WAS BLOCKED BY ANOTHER ACFT COMING UP ON FREQ AT THE SAME TIME. WE ASSUMED THAT SHE HAD CLRED US TO CROSS RWY 1R AND READ BACK TO HER '...CLRED TO CROSS RWY 1R.' HEARING NO OBJECTION OR 'HOLD POS' INSTRUCTION; WE TAXIED ACROSS RWY 1R. SHE THEN XMITTED; '...I TOLD YOU TO HOLD SHORT OF RWY 1R.' WE APOLOGIZED AND WERE CLRED TO TAXI TO THE E RAMP. WE; AS WELL AS THE TWR WERE AT FAULT IN THIS INCIDENT. WE SHOULD HAVE ASKED THE CTLR TO REPEAT HER INSTRUCTION BEFORE TAXIING ACROSS RWY 1R.