Check with the police department, truck stop, 24 hour restaurant, gym, fair grounds, church, school bus driver parking, motel truck parking area, hospital visitor parking or worse comes to worse drive around eat in a 24 hour restaurant, work it off in a 24 hour gym all night and sleep during the day in a day use area!
I am assuming that you're in a relatively nondescript vehicle. If you're in a big RV, I recommend joining Harvest Hosts. I've never been to Lincoln, NE. But I look for places to park in unfamiliar areas all the time.
Here are some places I'd check out if I were there:
The parking areas near the Museum of American Speed
Unrestricted parking lots around University of NE
Gateway mall
Lion's Gate
Lincolnshire Square Business Park
Park One
Hospitals- Bryan Medical, Lincoln Medical, St Elizabeth Hospital
Apartments- near 1806 Knox St, near 7221 Candletree Ln, Holmes Lake by Broadmoor, Lenox Village, Mercy Western Manor
If you're going to be in town for a while, its imperative that you move around. Don't stay in one spot for more than a night. Get up in the morning and move to a different spot if you need to hang out during the day.