DURING A NIGHT OP A B737 FLC FAILS TO FOLLOW TAXI INSTRUCTIONS FOR RWY 5L AT CLE ARPT. PIC TAXIES STRAIGHT AHEAD AT TXWY S AND STOPS ACFT WITH NOSE OVER THE HOLD SHORT LINE OF RWY 5R. CTLR NOTES THIS AND THEN ISSUES NEW TAXI CLRNC FOR ACFT.

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OPERATING AS FLT CLE-STL. FLT PUSHED FROM GATE AT CLE; WAS CLRED TO TAXI TO RWY 5L; BUT HOLD SHORT OF RWY 5R AT TXWY N. I DEPARTED THE RAMP ON TXWY S; BUT MISSED MY TURN ONTO TXWY L; THE PARALLEL. INSTEAD; I WENT STRAIGHT TOWARD RWY 5R. I REALIZED MY ERROR AND HAD STOPPED THE ACFT WITH THE NOSE MAYBE 2 FT ACROSS THE LINE WHEN THE GND/TWR CTLR SAID 'HOLD IT RIGHT THERE; ACR XY'. HE THEN CLRED THE ACFT LNDG ON RWY 5R TO SWITCH OVER TO RWY 5L. THE CTLR THEN ALLOWED US TO TAXI SW ON RWY 5R TO RWY 18 TO TXWY L; THEN TO RWY 5L AS PREVIOUSLY CLRED. I HAVE OPERATED IN AND OUT OF CLE SINCE WE STARTED SVC THERE. I KNEW BETTER THAN TO GO STRAIGHT ON TXWY S; BUT I THINK I WAS OVERLY TIRED DUE TO AN UNPLANNED; 3 HR GND HOLD AT THE BEGINNING OF OUR WORK DAY IN SLC. IN ADDITION; I MUST HAVE BEEN DISTR MOMENTARILY BY SOMETHING IN THE COCKPIT DURING TAXI. THE CTLR SEEMED UNCONCERNED; AND PROBABLY WILL NOT WRITE THIS ALL UP. WE THANKED HIM FOR A JOB WELL DONE. CALLBACK CONVERSATION WITH RPTR REVEALED THE FOLLOWING INFO: RPTR CALLED BACK AND STATED THAT THE CTLRS AT CLE WERE VERY ALERT. AS HE WAS BRAKING THE ACFT THE GND CTLR WAS TELLING HIM TO HOLD SHORT AND THE LCL CTLR WAS ALREADY SHIFTING THE LNDG TFC TO RWY 5L FROM RWY 5R. HE SAID THAT THE SIGNAGE COULD BE BETTER AT THAT AREA OF TXWY S AND TXWY L; SINCE IT IS QUITE A WIDE AREA. WHEN ASKED; HE STATED THAT HE WASN'T CERTAIN AS TO WHAT HIS DISTR WAS AT THE TIME BUT MOST LIKELY WAS SOME LEVEL OF INVOLVEMENT WITH THE CHKLIST AS THAT IS THE PLACE IT WOULD OCCUR.