I tried all the airports you listed, and this is the results:
VAAH - I got Celebi and Global Flight Handling. According to several sources, they both operates at VAAH
VABB - I got Air India, Bird India and Celebi
VOBL - I got GlobeGround India, Air India and Celebi
So, it seems to me the handlers are surely correct because, while in previous versions we might have included "all" airlines of India in a genetic rule for airports in India, sometimes we get suggestions from users that a certain ground handling company is on a specific airport, so that will override the generic rules, so you might think the other airlines "disappeared", when in fact we replaced the generic rule with more current information.
Of course, these are just the STOCK rules, but from the Operators tab of the Parking customization page, you can assign any operator you want, and those assigned by you, will take precedence over any rule.