I had a battery go bad in a fire alarm system. It has two 12 batteries in series to make 24 volts.

Because it was at night on a weekend, I took one battery out of a emergency light and installed it. The combined voltage was 24 volts but when connected and being charged the voltages were 14 volts across the charged battery and 10 volts across the discharged battery. So, I had to rob a second battery out of another emergency light.

It made me wonder:

If I install two new batteries and they do not have equal charge, like one being fully charged and the other discharged, will the charged battery prevent the discharged battery from charging?

Before I install the new batteries, would it make a difference if it let them sit in a parallel connection for a couple hours so the charges balance out?

Batteries connected is series have to be the same type and same state of charge. Virtually all batteries are cells wired in series but they were the same when it was new and theoretically they stay relatively the same. When one cell gets weak, the battery is toast. You are just seeing that effect with separate batteries.
There are troubling problems with batteries in parallel too although GM tried it with some of the Diesel cars. You pretty much had to replace them in matched pairs too.

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I have one of these with test leads attached, to test my inverter batteries at work. I clamp my Fluke 187 leads onto the lugs. A charged battery should still be above 12V with the 6 Amp load imposed. I'm referring to 7.5AHr & up batteries. It's not good to discharge your VRLA batteries below 1.75 Volts Per Cell(VPC). It's normal to see 14 or more volts across bad batteries on charge, and 10V or less when you disable the charger. With your FACP, it's supposed to run for 24 hours on standby and then 5 minutes in alarm. You should definitely swap them out together or you won't likely make the requirement. Remember that brand new batteries will self-discharge and die on the shelf, so you should float charge them or top them off at least every 6 months.
Joe