Newcastle struggled for the second week in a row to shake off Overmach Parma before holding out for a 21-16 win at Kingston Park.
Jamie Noon looked to have put the Falcons on course at the start of the second half with the Falcons' third try, following scores from Rob Miller and Micky Young.
But Parma replacement Juan Martin Soffredini crashed over on 58 minutes to salvage a losing bonus point for the Italians that leaves Pool Four in the balance.
Newcastle are now two points behind leaders Brive who eased to a comfortable victory at Cetransa El Salvador. Former England wing Ben Cohen clinched a bonus point with the fourth of Brive's nine tries in a 55-5 win in Spain.