You are not using Spreadsheet::ParseExcel to write - Spreadsheet::ParseExcel::SaveParser uses two different modules for read/write:

my $template = $parser->Parse("B_Mum_Busy_hour_report_fdb_$date.xls");
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reads entire Excel file into memory using Spreadsheet::ParseExcel

$template->SaveAs("B_Mum_Busy_hour_report_fdb_$date.xls");
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overwrites old Excel file using Spreadsheet::WriteExcel