FLC OF AN LGT FAILED TO MAINTAIN ASSIGNED APCH ALT DUE TO LACK OF HEARING AND UNDERSTANDING THE APCH CLRNC RESTRS.

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WX ON ATIS LOW VFR; BUT ILS ASSIGNED DUE TO CLOUD LAYERS. SNOW SHOWERS IN AREA WERE REDUCING VISIBILITY AND OBSCURING CLOUD DEFINITION LOWER AND CLOSER TO FIELD (NOT YET RPTED BY ATIS); BUT VISIBILITY OUT OF CLOUDS AND BTWN LAYERS GOOD FOR US. THEREFORE; THE CONVERGING TURNS ONTO PARALLEL APCHS EVOKED GREATER OUTSIDE AWARENESS AND LOOKOUT. IT WAS THE NOT UNUSUAL BARRAGE XMISSION FROM THE CTLR THAT PROBABLY WAS THE EVENT WHICH TRIPPED US UP. AN AFTER LNDG CALL TO RAPCON CONFIRMED. LOWERING VISIBILITY WAS DELAYING APCHING ACFT FROM RPTING 'FIELD IN SIGHT.' OUR CTLR WAS GETTING A LITTLE BEHIND AND GAVE A TURN; SPD CHANGE; ALT XING; FREQ CHANGE; APCH CLRNC AND CONTACT TWR AT; ALL IN THE SAME XMISSION; AND ABOUT THE SAME TIME THE TCASII GAVE A 'TA' FOR A PLANE JOINING THE PARALLEL APCH. BEING VMC AT THE TIME; I TOLD THE CAPT I HAD THE 2 O'CLOCK LOW TFC VISUALLY WHILE HE WAS READING BACK WHAT HE HEARD OF THE CTLR'S CLRNC. IT WAS A CLASSIC CASE WHEREIN WE SHOULD HAVE SAID: 'SAY AGAIN ALL AFTER (CALL SIGN).' TO TOP IT OFF THIS APCH USES AN OFF-FIELD DME SOURCE FOR ILS STEP-DOWN REF (IN ITSELF A RIDICULOUS CONDITION FOR A MAJOR INTL TERMINAL). BECAUSE I HEARD 'CLRED FOR THE APCH' THE 'CROSS AT' DIDN'T REGISTER SINCE A LOWER STEP WAS PERMISSIVE AT THAT DME. LESSON RELEARNED: REQUEST REPEAT; HURRIED OR NOT; IF THERE'S A QUESTION MARK OVER THE COCKPIT.