Star Alliance and Oneworld Alliance airlines attack Delta/SkyTeam with summer discount fares from mid $500s for San Francisco to Paris flights. Google Flights shows many low fare dates, but finding a direct booking link through Google Flights is a challenge. Search Momondo.com works for most dates where Google Flights does not offer direct links.
Tickets shown in this post were found through Google Flights direct links.
United Airlines/Star Alliance airlines and British Airways/Oneworld have San Francisco SFO to Paris CDG tickets for around $570 round trip for July 2018 departures.
British Airways has tickets around $600 round trip flying different oneworld Airlines.
SFO – Montreal YUL – Paris CDG – Frankfurt FRA – SFO
Tue July 17 – Thu July 26
British Airways SFO – Paris $606.87 round trip
SFO – Madrid MAD – Paris Orly ORY – Dublin DUB – SFO