ZOB CTLR; PROVIDING OJT; ISSUES B737 CONDITIONAL CLRNC THAT WAS NOT COMPLIED WITH; RESULTING IN A CONFLICT WITH PARALLEL DC9 TFC.

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B737 X TFC WAS GIVEN A HDG TO PARALLEL THE DC9 Y SO THAT THE B737 COULD DSND THROUGH THE ALT OF THE DC9 AND THEN TURN ON COURSE. THE B737 WAS GIVEN A CLRNC WHEN REACHING AN ALT CLR OF THE DC9 TO TURN ON COURSE. FROM THE READBACK WE RECEIVED; THE PLT UNDERSTOOD THIS BUT IN REALITY THE B737 PLT THOUGHT HE WAS CLRED ON COURSE BEFORE SEPARATION WAS ATTAINED. SUPPLEMENTAL INFO FROM ACN 457403: DSCNT ON WAKEN ARR; ZOB CLRED US TO DSND TO 16000 FT. WE WERE THEN CLRED OFF ARR WITH A 010 DEG HDG AND CONTINUED DSCNT. WE THEN DSNDED THROUGH FL180. WE WERE GIVEN A PARTIALLY BLOCKED CLRNC TO RETURN BACK DIRECT WAKEM. THE CAPT INFORMED - WE ONLY RECEIVED BACK TO DIRECT WAKEM AND WE MISSED THE FIRST PART DUE TO A BLOCKED XMISSION. THE CTLR REPLIED DIRECT WAKEM CLRNC CORRECT. THE CAPT READ BACK DIRECT WAKEM AND I TURNED THE ACFT BACK 30 OR 40 DEGS BACK TO WAKEM AND CONTINUED THE DSCNT TO 16000 FT. A MIN OR SO LATER; THE CTLR ASKED US TO VERIFY THE 010 DEG HDG. THE CAPT INFORMED HER WE WERE DIRECT WAKEM AS CLRED BY HER. SHE GAVE US AN IMMEDIATE L TURN BACK TO A 010 DEG HDG. SHE INFORMED US THE CLRNC WAS DIRECT WAKEM REACHING 16000 FT; A CLRNC THAT WE NEVER HEARD OR READ BACK. I OBSERVED A DC9; 800-900 FT OFF OUR R WING AT 3-4 MI; OBVIOUSLY THE TFC WE WERE VECTORED AWAY AND CLRED BELOW. AFTER LNDG; THE CAPT CALLED ZOB. THEY HAD ALREADY PULLED THE TAPES AND INFORMED THE CAPT THAT THE CTLR WAS AT FAULT FOR NOT GIVING US PROPER INSTRUCTIONS.