It won't notice 'Valentine's' as a path that needs to be created,

That's not what I see. I see it creating path "Valentine's Day" twice and path "Valentine's" twice.

Valentine's Day
Valentine's Day created new path!
Valentine's Day|Cards
Valentine's Day created new path!
Valentine's Day|Flowers
Valentine's     created new path!
Valentine's|Candy
Valentine's     created new path!
Valentine's|Telegrams
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For some reasons, you commented out use warnings; which tells you why you're creating the same path more than once.

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By the way,

$prev_path !~ /^$parent$/
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can fail since $parent contains plain text, not a regex pattern. You want

$prev_path !~ /^\Q$parent\E$/
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or just

$prev_path ne $parent
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