In this talk, I will describe an upper tail functional large deviation principle (LDP) for the parabolic Airy process, one of the universal objects in the KPZ universality class. Thanks to the recent metric level LDP for the directed landscape from Das, Dauvergne, and Virág, the LDP rate function for the parabolic Airy process can be expressed in terms of a variational problem. I will describe how to solve this variational problem by connecting it to the weak solutions of Burgers' equation. This talk is based on joint work with Li-Cheng Tsai.