Since you asked for a push, here's some pseudo-code, which you should be able to translate into perl:

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]

I expect others will propose more elegant and compact methods, but this is, I think, easy to comprehend. Note that watching for the "machine" name involves a regex match with the "i" modifier at the end, to ignore case. When expressed in perl, this approach can involve less typing than the pseudo-code I gave you.