I am a US citizen born in SF . Both of my parents were born in Italy. My father fought with the Italian army in WWI. I don't think they maintained their Italian citizenship after they were naturalized in the US. Can I obtain Italian citizenship?

You can and you should. The law says you can obtain Italian citizenship if at least one of your parents was Italian citizens when you were born. You can actually obtain it via your grandparents. The process in not even excessively long. Read this for the alternatives: https://www.icapbridging2worlds.com/italian-dual-citizenship-services/jus-sanguinis/

I have had an Italian ID card since the 1960's. When I went to get my Italian passport in 2018, I had to prove that my father was an Italian citizen when I was born, Relatively easy as I was in Italy and only had to go to the Commune where my father was born and get his Atto Di Nascita. In Italy the Birth Certificate has more than the date of birth, it includes the date of marriage, naturalization to US citizen and death. Since my father was an Italian citizen when I was born, I received a passport. Do you have your parents naturalization papers? You may be able to get copies through the Freedom Of Information Act. No background check for me.

Yes you can but it is a lengthy process. You will have to start by retriving both your parents birth certificates, then the consulate will instruct you with all other docs required. My friend did it it took almost 2 years. Background checks for you will be required too