This article is the first in the literature to investigate the network traffic equilibrium for traveling and parking with autonomous vehicles (AVs) under a fully automated traffic environment. Given that AVs can drop off the travelers at their destinations and then drive to the parking spaces by themselves, we introduce the joint equilibrium of AV route choice and parking location choice, and develop a variational inequality (VI)-based formulation for the proposed equilibrium. We prove the equivalence between the proposed VI model and the defined equilibrium conditions. We also show that the link flow solution at equilibrium is unique, even though both the route choices and parking choices are endogenous when human-occupied AV trips (from origin to destination) and empty AV trips (from destination to parking) are interacting with each other on the same network. We then develop a solution methodology based on the parking-route choice structure, where we adjust parking choices in the upper level and route choices in the lower level. Numerical analysis is conducted to explore insights from the introduced modeling framework for AV network equilibrium. The results reveal the significant difference in network equilibrium flows between the AV and non-AV situations. The results also indicate the sensitivity of the AV traffic pattern to different factors, such as value of time, parking pricing, and supply. The proposed approach provides a critical modeling device for studying the traffic equilibrium under AV behavior patterns, which can be used for the assessment of parking policies and infrastructure development in the future era of AVs.