The more I think about it, the more I think it just wouldn't work. I doubt the input pin is going to provide enough current to power your chip -- you'll get a stuck closed reading in input test and your circuit won't run because its mere presence pulled down the input pin to next to no voltage.
Adding a power wire up to the CP is the way to go.
No, the input pin isn't an issue. Usually these things are done with a resistor to +5 pulling up the line, and a CPU input. Drawing current on that line (or frankly, even driving that line at +5) is NOT going to damage the inputs - the input isn't sourcing any current. The pull-up may be on the chip, or separate and on the board), but the actual input line isn't doing the work.
The thing is that if you pull any significant amount of current you WILL lower the voltage, and there's a good chance that you'll take it down far enough that it will make the input unstable.
Here's a better question: does whatever you are doing have to work when the buttons are NOT pressed? Because if it doesn't, then you can hook it to the ground side of the buttons, using the buttons as +5 when they are pressed. If it's an auto-fire or something, then it could pick up current from the button and start working as soon as the button is pressed. Using a capacitor it could hold enough charge to make it through whatever cycle it's doing.
Good luck.
EDIT: looking at the Missile Command schematics, they use 470 ohm resistors on the button inputs. 470 ohms, 5 volts V=IR thus I=V/R means that you can get a maximum of 0.01064 amps (= 10.64 milliamps) through that line (and that WILL trigger the input). If you needed to parasite without triggering the input, you'd need to draw a lot less, perhaps 2 milliamps (that's a TOTAL guess, you'd have to play with it). Not saying it can't be done, but it doesn't look at all easy.