One of the most novel and powerful emerging nanotechnologies is quantum computing. This paper proposes a design for a parallel random-access memory (RAM) using Quantum-Dot Cellular Automata (QCA). QCA has the potential of becoming the technology of choice at the nano-scale. The RAM design is based on a simple 2-dimensional grid of identical memory cells, addressed by row using a QCA decoder. The proposed memory has storage capacity > 1.6 Gbit/cm2.