Random access coding is an information task that has been extensively studied and found many applications in quantum information. In this scenario, Alice receives an n-bit string x, and wishes to encode x into a quantum state ρ x , such that Bob, when receiving the state ρ x , can choose any bit i ∈ [n] and recover the input bit x i with high probability. Here we study a variant called parity-oblivious random access codes, where we impose the cryptographic property that Bob cannot infer any information about the parity of any subset of bits of the input, apart from the single bits x i . We provide the optimal quantum parity-oblivious random access codes and show that they are asymp-totically better than the optimal classical ones. For this, we relate such encodings to a non-local game and provide tight bounds for the success probability of the non-local game via semidefinite program-ming. We also extend the well-known quantum random access codes for encoding 2 or 3 classical bits into a single qubit. Our results provide a large non-contextuality inequality violation and resolve the main open problem in a work of Spekkens, Buzacott, Keehn, Toner, and Pryde (2009).