Once you cross the Line Drive bridge (over Salt Creek) it becomes clear where you're going to park. There are 2000 parking spaces organized in several lots around the Haymarket complex. Also there are many helpful parking attendants to direct you to a spot. The circle at the end of Line Drive is perfect for dropping people off or picking up tickets.
The south side of the complex has the largest parking lots and easy access to the berms from the secondary entrance. The north lots are smaller and closer to the softball field.
Allow extra time for parking since, as game time approaches, "Lincoln's Finest" arrive to deal with the "bottleneck" that the bridge becomes. Both lanes become "inbound" and your choice of parking is set by the direction you arrive --and how you'll be allowed to leave when the game is over.
Traffic approaching from the south on Sun Valley Blvd. is directed to the south lots and will be waved in that direction. Fans arriving from the north will be directed into the smaller north lots and north when leaving. If you're staying downtown you'd be smart to take the route on O Street from the south so you can return the same way.
After big games, especially fireworks nights, traffic is often clogged leaving the park. Between the police at the traffic light and the parking attendants it moves quickly (and continuously) compared to other parks, but be prepared for delays when leaving the ballpark.
If your plans are to go to the Historic Haymarket after the game, for food and drink or to return to your hotel, you might consider the Haymarket Connector pedestrian option.
Tailgating is allowed in the parking lots but there was little sign of it during my first visit. Part of the problem is the incomplete nature of the grounds. In the future, especially when construction is complete, there may be more people than the family I saw eating watermelon after a hot Sunday day game.
But there are clear issues that have to be addressed starting with an explicit "No Re-entry" policy and the absence of portapotties. Saltdogs management may find (as the Saints have) that tailgating works best if fans can return to the parking lot after getting their "free stuff." And if folks are eating they'll need access to restrooms. Something that might also be solved with allowing re-entry.