One of Parks & Recreation's running jokes is Leslie's deep affection for Vice President Joe Biden. She doesn't hold all politicians in such esteem, so there's something about Biden in specific. Has this ever been explained?
After fairly extensive researching, I can't find a rock-solid character-based reason or a quote from one of the creatives explaining why they chose Joe Biden as her celebrity crush specifically.
However, Joe Biden's everyman-quality and mix of enthusiasm, goofiness and predilection to gaffes makes him pretty similar to Leslie Knope. He makes sense as a hero/crush.
The reason that makes the most sense though is that it's funny. Having her be obsessed with him, the extreme sexual attraction and professional admiration don't really seem warranted to the average viewer. It makes her level of obsession even more strange and alien and funny. If she felt the same way about Obama or Hillary Clinton, the audience could be thinking, "oh, I see that, Obama actually is pretty hot for a president" or "damn straight, Hillary is a good role model and admirable politician".
Finally, Joe Biden has a more silly-looking face than the average politician and is famous enough to be funny to everyone without being controversial enough to turn off half the audience every time he's joked about.