Autonomous robots are expected to perform a wide range of sophisticated tasks in complex, unknown environments. However, available onboard computing capabilities and algorithms represent a considerable obstacle to reaching higher levels of autonomy, especially as robots get smaller and the end of Moore's law approaches. Here, we argue that inspiration from insect intelligence is a promising alternative to classic methods in robotics for the artificial intelligence (AI) needed for the autonomy of small, mobile robots. The advantage of insect intelligence stems from its resource efficiency (or parsimony) especially in terms of power and mass. First, we discuss the main aspects of insect intelligence underlying this parsimony: embodiment, sensory-motor coordination, and swarming. Then, we take stock of where insect-inspired AI stands as an alternative to other approaches to important robotic tasks such as navigation and identify open challenges on the road to its more widespread adoption. Last, we reflect on the types of processors that are suitable for implementing insect-inspired AI, from more traditional ones such as microcontrollers and field-programmable gate arrays to unconventional neuromorphic processors. We argue that even for neuromorphic processors, one should not simply apply existing AI algorithms but exploit insights from natural insect intelligence to get maximally efficient AI for robot autonomy.