It seems to me that parsing into individual emails won't really buy you anything. A simple parse is best. Here I illustrate the idea for id, path and size:

my %rec;
my $pid;
while (<>) {
   $pid = $1 if /^Backup ID: (\d+)$/;
   $rec{$pid}{path} = $1 if /^Path:   (\S+)$/;
   $rec{$pid}{size} = $1 if /^Size:   (\d+)$/;
}
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Because the attributes always come after the primary id, little bookkeeping is required to populate a hash, or a database. It isn't clear to me how the subject enters the problem, so I left that out.