High-T c superconducting coils wound by REBCO coated conductors with no turn-to-turn insulation have been widely used for high-field DC magnets, due to the fact that they have a higher engineering current density and can be self-protective during a quench. One long-lasting problem of no-insulation (NI) coils is the charging delay due to low characteristic resistance. In this paper, the authors propose the structure of a parallel co-wound NI pancake coil. Unlike a traditional NI pancake coil which is wound by a single tape, in the new design multiple NI tapes are co-wound. From a circuit point of view, the tapes are connected in parallel. It is discovered that the charging delay of the NI coil can be substantially reduced by the parallel co-wound method. The mechanism is that the inductance of a coil is proportional to the square of the number turns which is inversely proportional to the number of co-wound layers, whereas the characteristic resistance is nearly irrelevant to the number of layers, therefore the coil's time constant can be considerably reduced. The new design could also further increase the engineering current density of a magnet due to the independent current transporting of co-wound layers, which will be investigated in future work.