Advancing Tribal Parity on Public Land Act This bill addresses tribal interests in the disposal and management of public land. Specifically, the bill prohibits the federal government from disposing of public land or National Forest System land unless the Department of the Interior or the Department of Agriculture (USDA) determines, through consultation with any interested Indian tribe, that such disposal would not impact the rights and interests of any interested tribe and would not impair access to a reservation. Interested Indian tribe means an Indian tribe with (1) historic, precontact, cultural, or religious connection to a cultural site located on the tract of public land; (2) a former reservation located on the tract of public land; or (3) treaty rights or other reserved rights associated with the tract of public land. Interior and USDA must, prior to conducting a sale of a tract of public land, notify all tribes of the availability of land for sale. Further, Interior and USDA must sell the tract of land to an interested Indian tribe that submits a bid at fair market value. Land acquired by an interested tribe shall be taken into trust by Interior for the benefit of the tribe. Additionally, the bill revises various public land provisions, including to (1) add the interests of tribes to the list of considerations for land exchanges, and (2) authorize tribes to acquire land for recreational and other public purposes. The bill also requires each public land advisory board to include at least one representative of an interested Indian tribe.