It would be a good idea to link the grounds of the two regulator PCBs to avoid a DC offset, and of course paralleling diodes is not the best of ideas, as you certainly wont get their full rating, but if you must use those PCBs, minimally modified, its the best option. A better idea would be to strip both boards of diodes, replacing those on the 12V PCB with 3A Schottky diodes for less voltage drop - you'll probably need to drill out the holes in the pads and mount them vertically to get them to fit, and on the 5V board, replacing D2 and D3 with wire links to convert it to DC input, from the C.T. and the 12V Gnd respectively.
A center tapped secondary, two diode full wave rectifier has half the voltage drop of a full bridge, at the expense of much less efficient transformer usage. However even with Schottky diodes its likely to be marginal feeding a MC7805 from a 12V CT secondary. If you substitute a LDO, you *MAY* be able to get enough headroom (depending on the load current and the value of C)1 for a reasonable output current.