Popular child psychologists have advocated that parents

Hi mate.

Popular child psychologists have advocated that parents discipline male children similarly to the fashion in which they discipline daughters.

(A) similarly to the fashion in which they discipline
Wrong. comparison "male children" vs "the fashion" (parents discipline X (male children) similarly to Y (the fashion) <== wrong)

(B) in the same manner that they would use with
Wrong. comparison grammar. "in the same X that ...." is wrong. The correct structure should be "in the same manner as ....."

(C) like they would handle
Wrong. "Like" only + noun/noun phrase. [Like + clause] is always wrong.

(D) as they discipline
Wrong.
- If you pick D, you assume EVERY parents have both boy and girl. That's wrong meaning. The correct meaning is that Parents discipline boys as they would (unsure future) discipline girls (if they have).
- Good writing usually avoids repeating main verb (discipline) in one sentence. If the verb is the same, you should omit it, instead you should use auxiliary verb (do/did/will/would).

(E) as they would
Correct.
- (As + clause) is correct grammar.
- E conveys good meaning; in addition, E avoids repeating main verb (discipline). The structure of E is: parents discipline male children as they would (discipline) girls.

Hope it's clear.