My parents are flying from India to China. They have a newly purchased wrist watch to carry. Is it allowed to put the watch in either checked-in or carry-on baggage?
They can put the watch wherever they want. Watches are not safety relevant items (like laptop batteries) which can not be put into checked luggage.
Your parents may have to declare the watch at customs and may have to pay duty or taxes on it. But that's completely independent of what luggage they are stowed in. If it needs to be declared, custom will most likely want to see it, in which case it's easier if it's in a place that you can get to quickly.