AFTER LNDG SJC ON RWY 12R; A B737 FLT CREW ARE CLRED TO CROSS RWY 12L; BUT STOP JUST PAST THE HOLD LINES WHEN THEY NOTICE AN ACR ON TKOF ROLL.

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AFTER LNDG RWY 12R AT SJC AND TURNING OFF AT TXWY F; TWR GAVE A CLRNC TO 'CROSS RWY 12L AND TAXI VIA TXWY Y AND TXWY J TO THE GATE.' AS CAPT; I CALLED FOR ROUTINE 'FLAPS UP' DUTIES AS WE PROCEEDED TOWARD RWY 12L. WE BOTH VERBALLY CONFIRMED 'CLRED TO CROSS' AND BOTH WERE CLRING DOWN RWY WHEN I NOTICED AN ACFT ON THE TKOF ROLL ON RWY 12L. IN ALL THE RWY LIGHTS AND RAIN WITH THE WIPERS RUNNING ON INTERMITTENT; IT TOOK A SECOND TO DISCERN THE ACFT AND THEN REALIZE THE PROB AND I THINK I VERBALLY SAID; 'THIS GUY'S ROLLING' AND MADE AN AGGRESSIVE STOP ABOUT 5-10 FT OVER THE RWY 12L HOLD SHORT LINE AND ABOUT 30 FT SHORT OF THE ACTUAL RWY. JUST AS WE STOPPED; SJC TWR CTLR SAID 'XXX [FLT NUMBER] HOLD POS' WHICH WE ACKNOWLEDGED. THE TKOF ACFT WAS AN AIRLINER THAT WE NOW UNDERSTAND WAS ACR Y. WE NEVER HEARD HIS TKOF CLRNC NOR WERE WE TOLD HE WAS ON THE RWY. ACR Y WAS AIRBORNE AND AT AN ALT OF ABOUT 200-300 FT WHEN HE WENT BY OUR HOLDING POS. AFTER THE EVENT AND FURTHER TAXI CLRNC TO THE GATE; WE ASKED FOR A PHONE NUMBER TO CALL THE TWR AND DISCUSS THE EVENT. AFTER ARR AT THE GATE; I (AS CAPT) CALLED THE NUMBER AND TALKED TO THE CTLR NAMED XXXXX THAT ADMITTED IT WAS HIS FAULT AND THAT IT WAS A BAD CLRNC AND WAS DISCUSSING IT WITH HIS SUPVR AND DOING THE APPROPRIATE PAPERWORK; AND I TOLD HIM I WOULD BE DOING MY REQUIRED RPTS.