FO CLBED ABOVE ASSIGNED ALT DUE TO MISUNDERSTANDING ATC INSTRUCTIONS TO TURN TO A NEW HDG.

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DEPARTING IAH WITH TSTM ACTIVITY IN THE AREA. DEP CTL LEVELED US AT 11000 FT MSL. WE ASKED FOR DEVS AROUND A SMALL AREA OF CLOUDS (DUE TO EMER AIRWORTHINESS DIRECTIVE ON ACFT TYPE). ATC TOLD US TO DEVIATE AS NECESSARY THEN TURN DIRECT BPT WHEN ABLE. QUESTIONING THIS HE REALIZED WE WERE DESTINED FOR HRL AND THEN CLRED US RADAR VECTORS (170 DEGS) FOR WX. A COUPLE OF MINS LATER ATC TOLD US 'TURN R 210.' I ACKNOWLEDGED 210 DEG HDG. FO ASKED IF IT WAS HDG 210 DEG OR ALT 210. WHAT I THOUGHT I HEARD HIM ASK WAS '210?' WHILE MY HEAD WAS DOWN LOOKING AT AND ADJUSTING RADAR; FO INITIATED CLB. AT 12800 FT MSL; I NOTICE ALT ALERTER SET FOR 21000 FT. SUDDENLY REALIZING WE WERE CLBING OUT OF 12800 FT ON A 170 DEG HDG I LEVELED ACFT. ATC CALLED AT THIS SAME TIME AND MENTIONED WE WERE CLRED TO MAINTAIN 11000 FT. HE THEN CLRED US TO MAINTAIN 14000 FT; HDG 210 DEGS. POSSIBLE FACTORS IN THIS OCCURRENCE -- ACFT HAD EMER AIRWORTHINESS DIRECTIVE REQUIRING US TO STAY CLR OF ICING. TOTAL AIR TEMP MORE THAN 7 DEG C AT THIS POINT. DUE TO TSTMS IN AREA WE WERE BEING VECTORED AWAY FROM CLR AREA AND AWAY FROM PLANNED RTE. MOST ACFT BEING VECTORED THIS NEW RTE BUT ATC NOT TELLING ANYONE IN ADVANCE. FLYING WITH RELATIVELY NEW FO. FO AND I DID NOT COMMUNICATE PROPERLY TO CLARIFY WHETHER CLRNC WAS FOR CLB OR TURN.