DC9 CARGO FLC QUESTIONED ABOUT RWY XING AT JFK.

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WE LANDED ON RWY 13R WITH CLRNC TO 'TAXI TO THE RAMP WITH ME.' HOWEVER; AN ACFT THAT LANDED BEHIND US WAS INSTRUCTED TO 'CONTACT GND.' OUR TAXI RTE REQUIRED US TO CROSS RWY 13L. AS WE APCHED RWY 13L; WE NOTICED ANOTHER ACFT XING THIS RWY. NOT BEING TOLD TO HOLD SHORT; WE ALSO TAXIED ACROSS RWY 13L. WE WERE LATER INFORMED BY THE CTLR THAT WE HAD CROSSED AN ACTIVE RWY. HE ALSO TOLD US THAT WE WERE INSTRUCTED TO CONTACT GND CTL. THIS WAS INCORRECT. IN ADDITION; HE TOLD US THAT LCL CTL TRIED TO CALL US. THIS WAS ALSO INCORRECT; AS WE MONITORED THE TWR'S XMISSIONS AND RESPONDED THE FIRST TIME TO THEIR CALL TO US IN THE GATE. SUPPLEMENTAL INFO FROM ACN 474858: AFTER WE CROSSED; THE TWR CALLED US ON TWR FREQ AND REQUESTED WE CALL HIM ON THE LAND LINE. THE TWR SUPVR SAID WE DID NOT CONTACT GND AS INSTRUCTED; THEN STATED THEY COULD NOT REACH US ON TWR FREQ AND THEY DID NOT GIVE US CLRNC TO THE RAMP. WE FELT IT WAS BETTER NOT TO DISAGREE HOWEVER WE FEEL THE COMPROMISE IN SAFETY LIES IN WHERE ONE OPERATOR TRIES TO MAINTAIN CTL OVER ALL FACILITIES EVEN DURING SLOW PERIODS. THERE WAS DEFINITELY A COM BREAKDOWN BTWN THE TWR AND GND OPERATOR WHEN THEY CAME ON DUTY.