Even though Carter had fewer quarters in his piggybank than did his br

ArupRS , just to add one detail to aragonn 's answer:

The construction that you thought was wrong is both correct and common.
The verbs do, have, and be can perform many roles in a sentence.
Those verbs get confusing.

Some form of the verb "to do" can substitute for any other verb in English except to be, and the first verb cannot be an auxiliary i.e., cannot be coupled with other verbs/verbals.

Correct answer A contains the verb did, which is parallel to had,
but the option may look strange unless we realize that had is a simple past tense main noun, not an auxiliary (not a helping verb).

• Had/have/has as a "main" verb can mean possess, own, experience, and more.
Had in this sentence is an active noun. Carter is doing something.

Correct: I had plans to travel from Paris to Florence, as did he.

-- The Oxford Dictionary Online, here contains seven definitions of have (or had, or has, etc.) as a main verb.
By "main verb" I mean a verb that stands on its own or a verb that is not an auxiliary (helping) verb.
The examples in Oxford are good.
-- Another definition can be found HERE.

Non-native speakers should study what have as a main verb can mean. Doing so will prevent a lot of confusion.

• When can did substitute for had? When had is the main verb.

Had in the cases below is a main verb that means to possess or own.
What is Carter's action? Possessing quarters.
Did can substitute for had because both are past tense.

-- Correct: Carter had fewer quarters than Clark did.
-- Correct: Carter had fewer quarters than did Clark.

• did cannot substitute for had if had is an auxiliary verb.
Had in the cases below is an auxiliary verb and cannot be replaced with a one-word "to do" verb (do, does, did).
-- Wrong: Carter had collected fewer quarters than Clark did.
-- Wrong: Carter has been collecting quarters longer than Clark does.

Finally, the construction that initially you did not like is fairly common on the GMAT.
The subject/verb inversion can look strange, as can the substitution of did for had, but both are correct.

Correct: Even though Carter had fewer quarters in his piggybank than did his brother Clark . . .