Parallel wiring battery and amplifier board to a 12VDC 1A power supply safe?
In general, yes this can be done, but charging batteries is not trivial so you have to be careful.  How do you know the amp requires 2A, is that based on a rating, or measurement?  Ratings are not necessarily actual values.  For a power supply, the rating may be 1A, while it safely can deliver 1.5A.  For the amp they may state it requires 2A while it will still run on 1.  So in this example it will work from that power supply.  In both cases they add a conservative safety margin.  For amps in particular it may be that it will draw 2A only when delivering full volume audio.  

So you probably should have a power supply that can provide enough power for the amp if you were running the amp off the supply only.  If you run it off the battery and supply in parallel, they may share the load.  As to your concern about the battery getting overloaded if the power supply has a high current rating, it doesn't quite work that way.  The amount of current is primarily determined by the voltage difference of the battery and charger.  If the supply is producing 12V and the battery is already at 12V, then it will not draw any current, or possibly a small trickle current, even if the supply is CAPABLE of providing  100A.  But if the battery voltage is low, the current it draws from the supply will increase, up to the limit of the supply, so in this case it may be an issue.  

I'm still not sure what your battery chemistry is, I'm guessing LiPo or NiMH, but they each have different particular ways of charging.  If it's lithium based you can usually apply a current-limited and voltage-limited supply and it's pretty easy to charge, whereas NiCad or NiMH need more particular sequencing and charge termination.  Either way I don't advise just connecting a supply to the battery until you are very sure about what you are doing.  Even Lithium batteries that are 'easy' to charge as I mentioned can turn on you violently if overcharged/overheated, etc., as you are probably aware from the Samsung issue.