Why does my penis hurt when i cough?

It should not if you are completely healthy. Penile pain can be a referred pain which means one structure in the pelvic area such as prostate, testicle, etc. can be the source of pain and straining during cough can elicit that pain. If the pain is persistent and bothersome, you should see a urologist.

Very unusual symptom. Try wearing a support under wear and see if that helps