There are a number of regex's that would work here. The basic idea is to replace stuff that looks like <p> or <Xp>with nothing via a match global.

#!/usr/bin/perl -w
use strict;

my $test = '<p>this is some<1p> paragraph</p>';

$test =~ s/<.?p>//g; #s/<.??p>//g; #also ok

print "$test\n";
__END__
prints:
this is some paragraph
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