Is this a correct design to face this?

Not in general. @ISA is a package variable, it isn't bound to a specific instance. So, if you do:

my $x = My->new(source=>"xml");
my $y = My->new(source=>"json");
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then now both $x and $y are instances of a class that inherits from My::JSON. If My::XML and My::JSON only contain _init, and this is only called from new, you're getting away with it, but if they also contain methods that will be called elsewhere from the program, behaviour may be unexpected.

Perhaps you can reverse the inheritance order:

package My::XML;
use My;
our @ISA = qw[My];

sub init {...}

1;

package My::JSON;
use My;
our @ISA = qw[My];

sub init {...}

1;

package My;

sub new {
    my $class = shift;
    my $self = bless {}, $class;
    $self->init;
    $self;
}

1;

package main;
use My::XML;
use My::JSON;

my $x = My::XML->new;
my $y = My::JSON->new;
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