I replaced everything after the second open with this:

my %parents_of;
my $parentid;

while ( my $line = <FFINFO> ) {
    if ($line =~ /^Folder (\d{3,5})\s+\-\s+.*\./) {
        $parentid = $1;
    }
    elsif ($line =~ /\s+subfolder\s+(\S+)\s+\-\s+.*\./) {
        push @{ $parents_of{ $1 } }, $parentid;
    }
}

sub build_path {
    my $folderid = shift @_;
    my @parents = @{ $parents_of{ $folderid } || [] };
    return @parents ? [ map {
                                map { "$_/$folderid" } @{ build_path( 
+$_ ) }
                            } @parents ]
                    : [ $folderid ];
}

foreach my $k (sort (keys (%folders))) {
    $folderpaths{$k} = build_path($k);
    print "$k => $_\n" for @{ $folderpaths{$k} };
}
[download]

The highlights of the changes are:

• I read the subfolders file only once.
• I store the subfolders data in a %parents_of hash of arrays, which maps each folder ID to the folders that have it as a subfolder.
• In build_path, I use the hash of arrays instead of iterating over the whole subfolders file. It's still recursive.
• Since build_path returns an array reference now, the output loop has to iterate its contents.

For the ID that you pointed out, the new output is:

3053 => 100/3051/3053
3053 => 100/3057/3053
3053 => 100/3063/3053
3053 => 100/3066/3053
[download]

The new code produces all the output of the old code plus another 1108 lines, and it runs in less than 1 second while the original takes about 220 seconds.

I hope this helps.