Note that we only have two parallels here: "..to move.. to apply". The portion that comes after the underlined portion simply modifies "apply", don't get confused by the addition of "TO a purchase", this does not indicate that this is part of the parallel. Even at that, if purchase was parallel, then it would already be preceded by "and". It's not preceded by and, so that indicates that we have two parallels, not three. Thus, we only need "and" between the parallels
A) "programs that enable a family... to be able to move.. and to apply part of the rent..." is parallel, but the bolded portion is somewhat superfluous and distorts the meaning of the author. These programs do not unlock the ability to move, families have the actual ability to move even prior to the program. The addition of the infinitive "to be able" distorts the intended meaning
B) The only difference between A and B is that B omits "to be able". Much more concise and does not distort the intended meaning
C) subject verb agreement is violated (programs.. enables), also the semicolon creates a fragment. The parallel is 100% correct, but C is wrong for the aforementioned reasons.
D) subject verb error, (programs.. enables), and the parallel is erroneous because we have a present participle that is not in accordance with the infinitive "to move". D is wrong
E) the comma is weird, the "which" creates a inessential clause but the information we are given IS essential, it specifies something crucial about these programs, thus we need "that" instead of "which". Also, the parallelism is violated since we have the infinitive to move and the present participle applying.
So, B is correct!