Open the two files on different file descriptors. Redirect the input of the read built-in to the descriptor that the file you want is connected to. In bash/ksh/zsh, you can write read -u 3 instead of read <&3.
while IFS= read -r lineA && IFS= read -r lineB <&3; do
echo "$lineA"; echo "$lineB"
done <fileA 3<fileB
This snippet stops when the shortest file has been processed. See Reading two files into an IFS while loop -- Is there a way to get a zero diff result in this case?Reading two files into an IFS while loop -- Is there a way to get a zero diff result in this case? if you want to keep processing until the end of both files.
See also When would you use an additional file descriptor?When would you use an additional file descriptor? for additional information on file descriptors, and Why is `while IFS= read` used so often, instead of `IFS=; while read..`?Why is `while IFS= read` used so often, instead of `IFS=; while read..`? for an explanation of IFS= read -r.