Let me start by saying, unless you added living microbes to cycle the tank, your tank will not naturally cycle in 10-14 days. The cycling process doesn't start until you add an ammonia producer into the tank. In a natural cycle, it usually takes a couple of months or longer to complete. With using established filter material or " bacteria in a bottle", the cycling process gets down to weeks, but not days. This is a picture of the whole cycling process in graph form. You'll see how ammonia looks the same in the beginning and the end of that cycle. The Nitrite cycle will also look the same only there will be an increase in nitrates as the nitrites fall back to zero.
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The reason this is important is because adding more fish too soon only jeopardizes all the fish sooner. The only time it's safe to add fish is before the ammonia level starts to rise or after the nitrite cycle goes back down to zero. When you do a " fish in" cycle ( using fish as the ammonia producer) , you need to test regularly so that you don't allow the ammonia and nitrite level combined to go above 0.5 ppm. At that point you will need to be doing water changes to keep the levels below that combined 0.5 ppm in order to keep the water safe for the fish.
As for your current situation, you will not spread out the aggression with more fish. You will only add another fish for the Parrot to beat up on. There is obviously not enough hiding spots for the 3 fish or the Parrot has established a bigger area as HIS territory so any fish that comes into it is going to get some pushback. You COULD try rearranging all the decorations in the tank so that the 3 fish all have to establish new territories. No guarantees but this method does work with many territorial species.