Can a person take anticoagulants for pulmonary embolism even though aspirin makes their petechiae worse?

Yes. Aspirin is no treatment for pulmonary embolism. Anticoagulant like heparin and oral anticoagulant like warfarin or newer agents such as xarelto are required. There may be a problem if your petechiae are caused by low platelets but not if they are caused by vasculitis. A hematologist would collaborate with your physician to assure that you are properly treated or if anticoagulants can't be used.