artist,
First you need to understand that $self->param() ne $self->query()->param(). The former is more like a internal scratch pad available to all run-modes while the latter is the CGI query parameter method.

sub setup {
    my $self = shift;
    $self->param('id' => 123);
}
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Not knowing what you are trying to accomplish, you may also want to look into CGI::Session.