RWY INCURSION OCCURS WHEN AN SF340 CHK CAPT TAXIES ONTO RWY 24L AFTER CLRNC TO HOLD SHORT IN A NIGHT OP AT CLE; OH.

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WE LANDED ON RWY 24R FROM A VISUAL APCH TO CLE ARPT. I WAS GIVING IOE TO A NEW HIRE FO AND I HAD TO TAKE THE ACFT FROM HIM ON LNDG. AFTER TAXIING CLR OF RWY 24R; WE WERE CLRED TO TAXI; HOWEVER MY FO DID NOT GET THE CLRNC AND DID NOT UNDERSTAND THE INSTRUCTIONS; SO I HAD HIM ASK TWR TO REPEAT (THEY DID). WE WERE CLRED TO CROSS RWY 24C AND HOLD SHORT OF RWY 24L. THE LIGHTS FOR RWY 24C WERE TURNED OFF AND I WAS FOLLOWING BEHIND AN ACR Y ERJ. I FOLLOWED THE ERJ ACROSS THE FIRST RWY I SAW WHICH WAS THE LIT-UP RWY 24L (I DID NOT SEE RWY 24C). AN ACR Z B737 WAS IN POS ON RWY 24L AND I REALIZED MY MISTAKE WHEN I SAW HIS LNDG LIGHTS ILLUMINATED FOR TKOF. CONTRIBUTING FACTORS: I THOUGHT I WOULD HAVE SAME CLRNC AS THE ERJ AS I WAS CLOSE BEHIND HIM. RWY 24C WAS NOT ILLUMINATED (IT WAS CLOSED). THE TXWY LIGHTS ARE SPREAD OUT AND TEND TO BLEND IN WITH THE OTHER LIGHTS MAKING IT HARD TO IDENT THE TXWY LIGHTS. THE SAAB HAS A WEAK TAXI LIGHT. ALSO; 2 FREQS ARE USED; SO I DID NOT HEAR ANY CLRNCS FOR THE B737 JET. SUPPLEMENTAL INFO FROM ACN 600608: AFTER LNDG ON RWY 24R; OUR TAXI CLRNC WAS UNCLR. AFTER ASKING TWR TO REPEAT THE INSTRUCTIONS; TWR SAID TO 'CROSS RWY 24C AND HOLD SHORT OF RWY 24L.' WE WERE EXITING THE LNDG RWY USING TXWY R; WHEN THIS OCCURRENCE HAPPENED.