The Apollo Guidance Computer had fixed and erasable memory. The former (which is the focus of this question) contained instructions and constants, and was functionally equivalent to ROM. The latter was where data was stored, and was functionally equivalent to RAM. The response to parity errors in either memory was the same.
The effect of a failed memory bit is highly variable. In the worst cases, it could lock up the computer or branch to the wrong code. Most likely would be an incorrect computation with the computer otherwise proceeding normally. It's also possible that a bit error may be in code that would not be used, or in branch cases that did not get taken.
The AGC had several dozen programs, which the astronauts could select and run. The computer's response to a parity error was to reset the AGC to a state where the astronauts could select a new program (possibly even the same program they had been running). Unless absolutely needed to reset the computer to this state, data was preserved. So the astronauts could try the same program, a different program, manual control (if applicable), or (for the lunar module) use the Abort Guidance System instead.
Would the same problem recur? Again, it depends heavily on the location of the error bit. Some subroutines are used in virtually every program, and the problem would again manifest itself; other parts of the code execute under very specific circumstances and may never again be touched. In the worst case, there were procedures for dealing without the AGC, and the astronauts and ground crews practiced various contingencies in simulators before launch.
Although the computer was in the cabin, it was not practical to repair it during flight. Both types of memory were woven by hand under a microscope. The problem poses a broken wire, which would require unweaving and then re-weaving that line of memory. Even if the astronauts had the tools and materials available, they would be more likely to cause more damage or to get the pattern wrong.
Patching the code was not an option. The fixed memory was physically permanent. Even if you could write code to the erasable memory, there was no mechanism to execute code at an arbitrary address.
No parity error actually occurred during the entire Apollo Program. That link to Space.SE provides more information on parity errors.