Some experimentation with HTTP::Recorder suggests that what you want in place of $mech->click_button(value => "Fold it"); is instead: $mech->click('Action'); :

$agent->get('http://rna.tbi.univie.ac.at/cgi-bin/RNAfold.cgi');
$agent->form_number(1);
$agent->tick('toggles', '-noLP');
$agent->field('name', 'fakename');
$agent->tick('SVG', 'on');
$agent->field('email', name@domain.com');
$agent->field('Temp', '37');
$agent->tick('plot', 'on');
$agent->field('Params', 'RNA');
$agent->field('Sequence', 'GATTACAGATTACAGATTACA');
$agent->field('pffold', 'pf');
$agent->click('Action');
[download]

(HTTP::Recorder records scripts using "$agent" instead of "$mech".)

Viewing the HTML source of the page in question would have revealed your error:

<input type="hidden" name="rec-form1-submit-Action" value=1>
<input type="submit" name="Action" value="Fold it">
                           ^^^^^^
[download]

HTH,