There's a parking lot adjacent to the Swinging Bridge, though empty spots are a rarity in the summer - they usually only open up after a car has melted from exposure to the fumes wafting off the pit toilets here (the outhouses, thankfully, are out of range of the bridge itself). You have a slightly better chance of being able to park just past the bridge along the south edge of Sentinel Meadow. The bridge is also part of the Cook's/Sentinel Meadow Loop trail.
The Swinging Bridge isn't far from the lower falls trail (see the blue balloon on the map below), and you're unlikely to regret starting at the bridge and then hiking to the falls from there. Follow the trail on the other side of the bridge and you'll soon arrive at the Yosemite Lodge, which is just across Northside Drive from the trailhead.
View Swinging Bridge in a larger map