Lincoln THINKS they can ban legal CHP in the parks but it is only legal if they have posted conspicuous notice at every public entrance as per the definitions in the Administrative code laid out by the Nebraska State Patrol.
Otherwise the city of Lincoln has been stripped of the power to regulate transportation, possession or ownership of handguns carried under the rules and regulations of the Concealed handgun act.
Omaha has the same limitations, but I am not familiar with signage in Omaha parks