I asked my dad about this, here's what he had to say:
Won't burn out the transformer if the brown leads are disconnected. But you are correct that it's better to connect in parallel. Look carefully on the wires, and you'll see that the black wires are different, there is a plain black wire and a black wire with a red stripe. Likewise for the brown pair, a plain brown and a brown wire with red stripe. Oriented as the photo above the left wire is brown with the red stripe, then plain brown, followed by black with red stripe, and the most right wire is plain black. For 110 volts connect the first and third together (brown with red stripe and black with red stripe) and that is one side of the line cord. Then the second and fourth wires together (plain brown and plain black) to the other wire from the line cord.
For 230 volts, connect the left-most wire (brown with red stripe) to one side of the line cord, and the right-most wire to the other side of the line cord. Then join the center two wires (plain brown and black with red stripe) together as an isolated and insulated splice. This puts the two primary windings in series for the higher voltage option.
(note: it's really hard to see the red stripe on the brown wire, and fairy difficult to see it on the black wire, but they are there. if you can't see the stripes, then the order they come out is still valid, so the center two wires are the ones to connect for the isolated join)
