Can I safely and efficiently connect batteries parallel to a DC source?
That would not be feasible. Firstly, a 12V DC battery does not supply a consistent 12V. As the power gets drained off, the voltage drops. When the voltage supplied by the battery drops, there will be a potential difference between the two positive ends of the voltage sources and that would cause huge heat to dissipate through the wires. That would also mean that the wires would burn.