Neuromorphic computers could overcome efficiency bottlenecks inherent to conventional computing through parallel programming and readout of artificial neural network weights in a crossbar memory array. However, selective and linear weight updates and <10-nanoampere read currents are required for learning that surpasses conventional computing efficiency. We introduce an ionic floating-gate memory array based on a polymer redox transistor connected to a conductive-bridge memory (CBM). Selective and linear programming of a redox transistor array is executed in parallel by overcoming the bridging threshold voltage of the CBMs. Synaptic weight readout with currents <10 nanoamperes is achieved by diluting the conductive polymer with an insulator to decrease the conductance. The redox transistors endure >1 billion write-read operations and support >1-megahertz write-read frequencies.