According to Wikipedia, based on the 2014 estimate of ~850k people, the per square kilometer density figure for SF proper is 7,000.
Is there room for growth in SF? Definitely. You'll get no argument from me on that. And I agree with you that Parisian style density is a fantastic thing. It also goes to show you that skyscrapers aren't essential for high density urban living.
By the same token, I do think people tend to underestimate the density of 3-story tracts in SF. The zip code I'm living in in SF, for instance which is dotted with 2-4 story buildings for the most part boasts a respectable 31,000 people per sq mile (and this is on the west side of the city). Many observers, without actually looking at the actual population statistics, would probably say this is a low density neighborhood when the reality is that it is anything but.
If we can achieve those densities with a mere 3 stories, imagine how many more people we could house if we upzoned to 6.
Btw, you can find neighborhood density figures for the 2010 US census by zip code here: http://projects.nytimes.com/census/2010/map (look under the "more maps" drop down)
EDIT: Looks like you corrected the SF numbers.
* Barcelona: 16,000
* Buenos Aires: 14,000
* Central London: 13,000
* Manhattan: 25,846
* New York City (overall): 10,100
* Singapore: 7,600
* Paris: 22,000
* Central Tokyo: 14,500
Here's a map of the zoning ceilings throughout San Francisco: http://i.imgur.com/Tn7CSTX.jpg
Every yellow block in that picture is zoned 40-X, which means that buildings taller than four stories are not welcome. If we raised that to six stories (as is the norm, e.g., throughout most of Paris), we could easily accommodate hundreds of thousands of new residents.