I have tried an lot, but I can't create an working link with a email address different from the users email address that ist stored in a field.

Perhaps:
/pet/event_reminder?recipient=test.email.test&nid=348
/pet/event_reminder?mail=test.email.test&nid=348
/pet/event_reminder?to=test.email.test&nid=348

Nothing works.

Is this not indended to be done? But what ist this for:
"pet_to (required) - a valid recipient email address"?
This stands in the module line 384.
How can I use this in a block with php code?
Something like:
echo '<a href="' . $GLOBALS['base_url'].('/pet/test_email?pet_to&nid=%node:nid') . '">Test-Email</a>' ;