a pay deal [that] some claim would see them lose up to 20% of their salary
The italicized clause is a nested clause. The subordinate clause which acts as a modifier to the noun phrase also acts as a matrix clause which contains another subordinate clause. In the original, it is a contact clause, which simply means that it is a relative clause in which the optional relative "that" is omitted. The entire relative clause modifies "a pay deal".
Cast as an independent clause in canonical word order, the clause can be read as:
Some claim [that] that [pay deal] would see them lose up to 20% of their salary.
In this version of this matrix clause, we find a content clause acting as the direct object of the verb "claim". The subject of the content clause is not optional. In the original subordinate clause, that subject is a part of an object, which is the reason that it can be omitted.
The verb of this clause is "claim". Although "claim" also exists as a noun and the phrase "some claim" could represent a countable yet unspecified claim, the phrasing "some claim" in this sentence represents a subject/verb pairing: an unspecified number of people make a given assertion.
That would see them lose up to 20% of their salary.
Here, "them" refers to the subject of the outermost matrix clause, "teaching assistants in County Durham". "That" refers to the pay deal which the teaching assistants rejected. The sense of "to see" used in this clause is permissive: the deal would allow the assistants to lose a portion of their salary. "Them" is the direct object, and "[to] lose . . ." is the object complement.
The "would see" is in the subjunctive mode, which is appropriate both because the pay deal is contrary to fact (it was rejected) and because the content clause is the object of the verb "claim".
The levels of nesting in the original sentence go quite deep. "Would see" is the predicate of the clause that is the object of "claim". "Claim" is the predicate of the clause which modifies "a pay deal". That noun phrase is the direct object of the gerund "rejecting", which in turn is the object of the preposition "after". That prepositional phrase modifies the infinitive "to strike". The infinitive phrase is the direct object of the verb "voted". With an infinitive, a gerund and a finite verb between "voted" and "would see" and since the finite verb happens to look the same as a common noun, formally parsing this sentence does present a bit of a challenge.