While there isn't total consensus on the Padres being the Dodgers' primary rival — Max Muncy was adamant that you can only have one rival, and for the Dodgers it's the Giants — it's easily the prevailing sentiment, and you don't have to look any further than the two teams' regular season matchups this year to prove it.
The Dodgers-Padres rivalry is very clearly the most heated rivalry in baseball right now. There are others that have more history behind them, including Dodgers-Giants, but there's something different about LA's beef with San Diego, and vice versa.