The sentence is carelessly constructed, so it won't sustain a strict parse; but here's what's going on.  

*Over* here is attached to *contention*.  Verbs like *contend, quarrel, dispute, argue* often take a preposition phrase with *over* as a complement  designating the matter which is disputed:

> The sisters quarreled **over** who should get their mother's silver.  
> The king and his opponents contended **over** the rights of minorities and parlements.  


A noun derived from a verb often may take as a post-positioned modifier the same sort of preposition phrase which acts as the verb's complement.  

> [<sub>CLAUSE</sub>They contended **over** X.] &rarr; [<sub>NOUN PHRASE</sub>Their contention **over** X]

From there the attributive PP is easily recast as a predicative PP.  

> [<sub>NOUN PHRASE</sub>Their contention over X] &rarr; [<sub>CLAUSE</sub>Their contention **was** over X.]

And that finite clause may be recast as a non-finite clause so it can be subordinated:  

> [<sub>FINITE CLAUSE</sub>Their contention **was** over X.] &rarr; [<sub>NON-FINITE CLAUSE</sub>... their contention **being** over X.]

A participle clause like this may be employed as an *absolute construction*: 

> ... a grammatical construction standing apart from a normal or usual syntactical relation with other words or sentence elements. It can be a non-finite clause that is subordinate in form and modifies an entire sentence ... &mdash;[Wikipedia](https://en.wikipedia.org/wiki/Absolute_construction).

The participle clause headed by *being* modifies the main clause, supplying the matters ***over** which* the king's authority was challenged.

What throws a monkey wrench into this particular absolute construction is Dr. Linton's expansion of *contention* to *sources of contention*.

> The main sources of contention being over X ... 

Now the subject of *being* is no longer *contention* but *sources*; and it is not idiomatic to speak of a source as ***over** X*. The *sources* of contention, in the sense in which Dr. Linton uses *source*, is *X* itself. Dr. Linton should have written:  

> ... the main sources of contention being <s>over</s> the rights of religious minorities (Jansenists and Protestants) and the rights of the parlements.  

I hasten to add that this is a very minor error, which would pass entirely unnoticed in speech. Dr. Linton is a graceful writer; I imagine she rewrote this sentence many times, and somehow the *over* never got deleted. It's very easy to make this sort of error, and very hard to spot it.