The most common and most likely answer to this crossword clue is the 7 letter word REARING
I've seen this crossword clue in The Times
R
E
A
R
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More clues leading to the result REARING
Buddy explains!

The straight part
We know that "parents’ duties" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case parents’ duties = rearing.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 REARING mean?
REARING verb- To bring up to maturity, as offspring; to educate; to instruct; to foster.
- (said of people towards animals) To breed and raise.
- To rise up on the hind legs
- (usually with "up") To get angry.
- To rise high above, tower above.
- To raise physically or metaphorically; to lift up; to cause to rise, to elevate.
- To construct by building; to set up
- To raise spiritually; to lift up; to elevate morally.
- To lift and take up.
- To rouse; to strip up.
- To move; stir.
- (of geese) To carve.
- (regional, obsolete) To revive, bring to life, quicken. (only in the phrase, to rear to life)
- To place in the rear; to secure the rear of.
- To sodomize (perform anal sex)
- Act of raising young.