MULTIPLE RWY OP PARALLEL RWYS. ACR X HAD LTSS FROM MLG TKOF ON PARALLEL RWYS.

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ACR X DEPARTED DTW. THE TWR (LCL) GAVE A WDB THAT DEPARTED AHEAD OF US A 230 HDG AND TOLD HIM TO RPT ON THE HDG. TWR TOLD US THAT THE PRECEDING WDB WAS TURNING R AND CLRED US TO DEPART RWY 21R AND MAINTAIN RWY HDG. WE REPLIED THAT WE HAD THE WDB IN SIGHT; RWY HDG AND CLRED TO GO. UNKNOWN TO US; ANOTHER MLG WAS CLRED TO DEPART WITH RWY HDG INSTRUCTIONS AT THE SAME TIME FROM RWY 21C (ON TWR FREQ 118.4 -- WE WERE ON 135.0). TWR NEVER ASSIGNED US ANY OTHER HDG AND TURNED US OVER TO DEP. WE CHKED IN WITH DEP SAYING 'ACR X WITH YOU OUT OF 1200 FT CLBING TO 10000). HE LEFT US ON RWY HDG UNTIL APPROX OUT OF 3000 FT WHEN HE SAID 'TURN R IMMEDIATELY TO A HDG OF 260 DEGS.' WE HAD WATCHED THE OTHER MLG THROUGH THE CLB AS HE PARALLELED US BOTH VISUALLY AND ON TCASII. SUPPLEMENTAL INFO FROM ACN 242233: AFTER TKOF; MLG ON RWY 29C WAS LAUNCHED AT SAME TIME AS US; AND AFTER SWITCH TO DEP; WE NOTICED A POTENTIAL CONFLICT ON TCASII. DEP CTLR TOLD US TO FLY 260 DEGS IMMEDIATELY; THEN ASKED WHAT HDG WAS ASSIGNED BY TWR -- HDG WAS RWY HDG. SEPARATION WAS 1 MI AND 400 FT VERT WHEN 260 DEG HDG WAS ASSIGNED. I SUSPECT WE SHOULD HAVE BEEN ASSIGNED THE 230 DEG HDG; NOT THE RWY HDG WE FOLLOWED OFF RWY 21R.