The most common and most likely answer to this crossword clue is the 8 plus 8 letter word RELATIVE MAJORITY
I've seen this crossword clue in The Times
R
E
L
A
T
I
V
E
M
A
J
O
R
I
T
Y
Buddy explains!
The straight part
We know that "hallmark of largest party?" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case hallmark of largest party? = relative majority.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.