The program is around 10,000 lines and I don't know exactly which part of it is the culprit.

Perhaps a few debug statements with time stamps in a log file would help

sub logger{
$time=localtime();
print LOGFILE "$time : @_\n";
}
sub parseElement{
logger("I am in UR parseElement starting");
...
[download]

that way you should be able to isolate where the issue arises