I very rarely use Date::Parse or Date::Manip. The core module Time::Piece usually does everything I need. Here is an SSCCE showing it correctly parsing the two sample dates which you gave in your prose (but which do not appear in your code sample).

use strict;
use warnings;

use Time::Piece;
use Test::More tests => 6;

my $str = '31/08/2024';

my $tp = Time::Piece->strptime ($str, '%d/%m/%Y/');
is $tp->mday, 31;
is $tp->mon, 8;
is $tp->year, 2024;

$str = '01/09/2024';

$tp = Time::Piece->strptime ($str, '%d/%m/%Y/');
is $tp->mday, 1;
is $tp->mon, 9;
is $tp->year, 2024;
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If you want to persist with Date::Parse and/or Date::Manip that's fine but you will doubtless help others to help you by showing the problem as an SSCCE like this.