The most common and most likely answer to this crossword clue is the 11 letter word FAVOURITISM
I've seen this crossword clue in The Times
F
A
V
O
U
R
I
T
I
S
M
More clues leading to the result FAVOURITISM
Buddy explains!

The straight part
We know that "Having pets" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case having pets = favouritism.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.
What does FAVOURITISM mean?
FAVOURITISM noun- The unfair favouring of one person or group at the expense of another.