read first line from file_a
split the line into scalar variables ("machine" and "site")
open file_b
while reading a line at a time from file_b {
    if the line contains "machine", set $apply_edit to true
    elsif the line contains "Parent Submap: " and $apply_edit is true 
+{
        replace the rest of this line with "site"
        reset $apply_edit to false
    }
    print the line
}
[download]

But you can't just substitute arbitrary text in a text file, unless it happens to have exactly the same length. As file_b is being read and searched, each line should be copied out to a new file, and updated line printed in it's proper order, and the remainder of file_b copied. Then rename the new file to file_b (making sure file_b is closed).

Yes, I glossed over the output management part, thinking the OP probably had a handle on that already (unfortunate pun duly noted).

As often as not for things like this, I just make the script print to STDOUT and put the file handling stuff on the command line -- why right write more lines of perl code to do stuff that the shell can do for me with an angle bracket?

text-altering-script.perl  input.file > output.file

# optional step:
mv output.file input.file
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#!/usr/bin/perl -i.bak
  :
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$ ./myfilter -i file_a file_b
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while (<filea>) {
    if ( this is a line i want ) {
        s/[\015\012]+$//; # Just to remove any CR/LF
        my($router,$site)= split /,/,$_,3;
        $replace{$router}= $site;
    }
}
my $replacements= '\b('.join('|',keys %router).')\b';
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my $symbol;
while (<>) {
    if (my $hit/^SYMBOL: $replacements/i ... /^END SYMBOL/i) {
        if ($hit==1) {
            $symbol=$1;
        }
        else {
            s/^(Parent Submap:).*/$1 $replace{$symbol}/;
        }
    }
    print;
}
[download]