Phuket is one of the more manageable tropical destinations to bring parents to, provided you understand two things: there is no rail or metro of any kind, and the sea is not always safe.
On transport, the island runs on songthaews - pink ones circulating around Phuket Town, blue ones connecting the town out to the beach resorts. They're cheap and they're an experience, but they are open-sided trucks with a high step up and a bench along each side, which is awkward-to-impossible for anyone with a bad hip or knee. There is also the Phuket Smart Bus along the west coast and a scheduled bus between the airport and town, though routing and frequency are worth re-checking before you rely on either. With elderly parents, the honest answer is Grab, metered taxis or a hotel car, and just paying for it. The airport is 32 km north of Phuket City, at Mai Khao in Thalang district, so build in the better part of an hour of transfer at each end.
The water is the thing people underestimate. The whole west coast - Patong, Karon, Kata, Kamala, Bang Tao - faces the open Andaman and develops serious rip currents during the southwest monsoon, roughly May to October. The Government of Canada names Phuket specifically for strong riptides and notes deaths from drowning. Red flags on the beach are not advisory and are not about the weather being a bit grey. People drown here every year, often strong swimmers, often after ignoring a flag. If you're travelling with children or with anyone who isn't a confident swimmer, either come in the dry season from November to April, or plan around the calmer southern and eastern bays and the hotel pool. There are also jellyfish at some beaches at some times of year - ask locally rather than guessing.
On terrain: at beach level Phuket is flat and easy, and the resorts - particularly around Laguna, Bang Tao and Karon - are built with lifts, ramps and level ground in a way that generally works for wheelchairs, though it is worth confirming with the specific hotel rather than assuming. The temples don't. The Big Buddha and Wat Chalong both involve stairs, and the Big Buddha sits at the top of a long climbing road. Phuket Old Town is flat, but the covered five-foot walkways in front of the shophouses are narrow, uneven, and interrupted by a step up or down at every building line, so a wheelchair often ends up in the road. Promthep Cape has a paved terrace, which is fine, but the headland path beyond it is rough.
Boat days deserve a specific warning. A speedboat island-hopper means climbing in and out of a bouncing boat from a beach, several hours of hard slamming across swell, sun with no shade and no toilet. It's tiring for a fit adult and genuinely rough on an older parent or a small child. If you want a boat day with mixed ages, take a large ferry or a big-boat Phang Nga Bay trip on the sheltered east side rather than a speedboat to Phi Phi.
It is hot and humid year-round with no cool season - plan outdoor things for early morning and late afternoon and write off the middle of the day. Rabies is present in Thailand and dogs roam beaches and temple grounds, so keep children from approaching them. Do not rent a motorbike: the Government of Canada advises avoiding driving or riding motorcycles in Thailand at all, even for experienced motorcyclists, and notes that motorcycle accidents cause most road deaths there and that many rental bikes are poorly maintained. The one genuine reassurance is medical - Phuket City has several private hospitals used to treating foreign patients, which is more than most island destinations can say.