so how can I avoid having three separate csv parsing subs?

One way would be to build an iterator for each different input type and pass that to the parsing routine. Something like this:

use strict;
use warnings;
use Carp;
use Scalar::Util qw( openhandle );

sub filehandle_iterator {
    my $fh = shift;
    return unless openhandle( $fh );
    return sub { 
        my $row = <$fh>;
        chomp $row if defined($row);
        return $row;
    };
};

sub file_iterator {
    my $filename = shift;
    no warnings; # to avoid warnings about filenames with \n in
    return unless open my $fh, '<', $filename;
    return filehandle_iterator( $fh );
};

sub string_iterator {
    my $csv_string = shift;
    my @lines = split( /\n/, $csv_string );
    return sub { shift @lines };
};

sub parse_csv {
    my $csv_input = shift;
    my $row_iterator = 
        file_iterator( $csv_input )
        || filehandle_iterator( $csv_input )
        || string_iterator( $csv_input )
        || croak 'could not find iterator';
    while ( my $row = $row_iterator->() ) {
        # ... do stuff with row ...
        print "> $row <\n";
    };
};
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