A group of people specialized in the same field can be divided on particular issues. You may have the X (sub)group who think one way and the Y group who think another way. Then X and Y will be called "chapelles". And of course they will dispute, as josephboen said.
Literally, the idiom means "parish-pump quarrel" or in other words, a dispute within the same group, class, neighborhood etc.
>> This system would increase Perrin's power and that of his clique. More internal squabblings would ensue as well.
>> Perrin's influence ......... would be greater and there would be more internal disputes.
>> Perrin and his clique would be more influential, and there would be more disputes among them.