There are wireless doorbells available now, and that would save you the trouble of running wires through your house. My understanding is that wireless doorbells are battery operated, and that would also address your concern about power requirements as each door bell would have it's own power source.
But, just for the record, transformers CAN be connected in PARALLEL to provide additional power for door bells, zone valves and any other such low voltage AC equipment. Generally you can't do that if the transformers are of different impedance because the different impedance will result in one transformer's magnetic field developing slightly earlier or later than the other's, and that will result in a voltage difference between the transformers which will cause one transformer to drive a current through the secondary windings of another transformer. But, if you have multiple IDENTICAL transformers, you can connect them in parallel to double, triple or quadruple the power output from a single transformer. That's because if they're all identical and connected in parallel, then each one's magnetic field will develop at the same time from the applied AC voltage, so the voltage at all the secondary coils will be identical at any given time, and therefore there won't be any voltage difference between transformers to cause any current in any other transformer's secondary coil.
Heath Zenith | Wireless Battery Operated Door Chime Kit With White Cover | Home Depot Canada
Google Heath Zenith to find their web site and check to see if they sell additional chimes for this kit so that you can have multiple chimes sounding in different areas of your house if your door bell is pressed. I'd be surprised if they didn't as that would be the natural extension of a single battery operated wireless door chime.
Also, at least one company (Dimango) makes an "Extend-a-chime" door bell kit. This one requires that you connect a transmitter to the low voltage wiring of your existing chime. The transmitter then sends out a signal that causes all the other Extend-a-chime door bells in your house to sound.
That'll be 35 cents. Please contact the forum moderator to arrange for payment.