If I am a US citizen, can I petition for K-1 visa for my fiancé and immigration visa for my father at same time?

Assuming eligibility otherwise exists, a U.S. citizen may sponsor multiple people in immigration applications at the same time. In connection with the Affidavit of Support that must be included in application processes by which both your fiance and your father may seek to become Lawful Permanent Residents (to get "Green Cards"), your "family size" will need to contemplate both people, and if your documentable income may be insufficient to do so, then it may be necessary for you to have a joint sponsor in the application process. Generally a joint sponsor may be any U.S. citizen or Permanent Resident who has sufficient documentable income and who is willing to assume the responsibilities associated with the Affidavit of Support. Both a fiance petition (with a subsequent adjustment of status application) and consular processing or adjustment of status for a parent can be more complex than may be apparent just from reading the applications' instructions, the USCIS website, and the immigration statute. It would be wise for you to consult with an immigration attorney who, after learning all of the relevant information, could advise about eligibilities, options and strategies to attain your goals, and who then could offer legal representation in the often complex application process.