The Thousand Pound Petersons consist of Bryan (605 lbs) and Marla (395 lbs), a married couple who filed a lawsuit in 1982 against Pawnee. The fought for tax deductions based not on the number of children in a family, but by aggregate weight. At the time, the lawsuit appeared harmless, but it is said to be the cause of other legislation and services (such as a written law that incentivizes weight gain and two private hospitals dedicated to diabetes and bedsore treatment). The Petersons won the lawsuit.
The information on the Thousand Pound Petersons can be found on Leslie Knope's 2012 Congresswoman Campaign site, here.