。
Yumi and Emi were studying for their English test.
Literal
Yumi [and-と] Emi [topic-は] English-of test-[for-のために] study-[past-progressive-ていました].
と links two names conjunctively, and the topic marker は applies to the pair. のために marks the purpose/beneficiary: 'for the test.' 勉強していました is the past progressive — 'were studying.' Note how topic は scopes over the whole conjoined noun phrase Yumi と Emi.