The quickest way to do this is to start the game, then use the D-Pad menu to start your own game in Freeburn Online. Wait for someone to join, then use the D-Pad menu again to start a race. You don't even need to win! Just finish it.
However, if you're going for the completion, it's more efficient to combine this with the 'First Win' achievement for winning an online race with 8 players:
The best way to get this is through a boosting session. These appear regularly for Burnout Paradise, as people boost for wins and rivals. Check here:Burnout Paradise Remastered Sessions
If there isn't a session set up yet, you can make your own quickly by visiting a previous session and clicking the three dots in the top right, then 'clone session'
Eg clone this one: Boosting Gaming Session for Burnout Paradise Remastered
Make sure to set it a week or so in the future, so there's enough time for people to join.
PS - the quickest race you can create is a straight race down the wide North-South road in the East of the city, starting halfway down the map, and finishing outside the baseball stadium.