Apparatus and methods to support parallel access to services of multiple subscriber identity modules (SIMs) and/or electronic SIMs (eSIMs) by a wireless device are disclosed. The wireless device is a single-radio wireless device that includes wireless circuitry that supports an active voice or video connection via a single radio access network of a cellular wireless network at a time. To send or receive data for a second SIM/eSIM, when an active voice or video connection for a first SIM/eSIM uses the wireless circuitry, the single-radio wireless device can i) establish parallel, linked protocol data unit (PDU) sessions for different SIMs, ii) multiplex traffic for different SIMs over a common radio connection with SIM information embedded in protocol headers for routing by network equipment, and/or iii) use an additional dedicated bearer using a first, active SIM for each data network name (DNN) over which to route for a second, suspended SIM.