Overall, the two very different methods of transportation have surprisingly similar amounts of emissions. The exact circumstances make each better in some scenarios, but overall airplanes are slightly better for the environment for a 2-person trip.
The general question here about cars vs planes has been studied in great depth by experts, so I'll refer to them wherever I can. The EPA, for example, published a detailed report1 on vehicle efficiency that's regularly updated. I've reformatted a table from that report here and made the assumption that only two passengers are in the vehicle.
The GHG's 2018 GHG Conversion Factors for Company Reporting reports similar numbers but with a different methodology.
Your question asked specifically about an A320. According to the the GHG's report, an A320 is about average in load factors and emissions except for (rare) long-haul. So if you chose an A320 and not an A380, your long-haul numbers would be even better than suggested by the table above.
If, And's, and But's
There are lots of considerations here, like vehicle fuel efficiency, extra hotel visits, cargo, etc. However, we could just be getting nitpicky, so let's look at how big a factor those are. For example:
If you're in a car, your typical trip along some of the busiest airplane routes is about 1.08 to 1.27 times longer than a trip directly point-to-point (my own estimates)
An extra 50 lb. suitcase would require about 33 g CO2 per mile 1
An extra passenger in an automobile adds minimal extra emissions, but a lot of extra emissions on a plane
Cars produce several times more CH4 per passenger mile than airplanes do, and a little more N2O. By my estimates looking at  this makes cars ~15% worse than if the CO2 or gallons fuel burned is considered alone.
Car emissions can vary a lot, with a 1984 SUV producing 8 times as much CH4 and 20 times as much N2O compared to a recent sedan.1.
Getting to and from the airport might take an extra 20-60 miles of travel each way
Staying in a hotel room (you're not really going to drive from San Diego to Portland in one day, right?) creates an extra 15.13 kg CO2 2
If we graph these considerations together we can compare their size. These are rough estimates and your exact emissions will vary.
Some of these considerations are huge, although some other considerations like staying in a hotel room or the trip to and from the airport are pretty minor. It makes little sense to say air travel is better for the environment without adding that many automobile trips are, in fact, better because there are 3+ people travelling or the trip is short.
It's an apples to oranges comparison
However, just comparing emissions per mile between the two is a little misleading. There are some considerations like
- Automotive traffic contributes to smog
- Many people go places by plane they would never go to by car
- Air travel often requires other travel considerations at the end like not being able to travel as freely or in your favorite vehicle
Contrails' climate effect makes for a similar apples-to-oranges comparison. Contrails cause enough heating for their effect alone on global temperatures to be measurable. However, contrails also go away after just about a day, while CO2 emissions stay around for decades, some of it taking several millennia to dissipate.
1. "Emission Factors for Greenhouse Gas Inventories", EPA, last updated 9 March 2018
2. CHP in the Hotel and Casino
Market Sectors, EPA