If you're expecting to wash / play with a 'family-owned' elephant, perish the thought! Elephants are expensive creatures to care and feed for, and you aren't going to find personal families owning elephants.
Here's what you can do: go to a wildlife sanctuary / national park in India. Mark's answer mentions Jim Corbett National Park, which is in Northern India, close to the town of Nainital. Corbett National Park is actually best known for as a tiger reserve - part of a project known as Project Tiger to conserve tiger population in India. (As in aside, do read Man Eaters of Kumaon by Jim Corbett; it's a brilliant-if-dated book that took a first serious look at how human populations could live in harmony with tigers without harming their population.) Corbett National Park is one of few national parks with a sizeable elephant population, so that's one you can consider.
I gather from your other questions that you have Kerala in your itinerary. Each nature reserve in India has its own signature animal, and while Periyar Wildlife Sanctuary in Kerala is a technically a tiger reserve too it is best known as an elephant reserve. As part of organised tours you can book, you will be able to do the activities you're interested in. Obviously, tours like these can be somewhat pricey (this depends on what you consider "pricey") as the money goes towards conservation efforts - and is also somewhat touristy. At least you know what you're spending is going towards a good cause.
Another option is: go to a temple in South India! Elephants are worshipped in South Indian culture and many temples have their own stable of elephants that are worshipped, bathed, and fed daily. Now this isn't something that is organised so I don't know how you can confirm / ensure you get to do it (since the elephants are holy, many people want to do it), but temple devotees do do it (but then, as a non-Hindu I'm not sure whether you'll be allowed?). One of the temples particularly famous for its collection of elephants is Guruvayur Temple (again in Kerala). Regardless, a trip to see temple elephants is totally worth it!