What would it mean for disney company? Would disney keep naming rights or would it be verizon world? And Verizon's Cinderella movie etc?
The Disney name is worth FAR more than the Verizon name. Probably all you would see is a tag ling at the bottom of the traditional Disney logos that says "A Verizon Company"
There would be a few large risks with a buyout.
1) They could spin off the parks like In-Bev did with SeaWorld and Busch Gardens. I see about zero percent chance of this happening. The parks are a huge percentage of Disney's revenue and name recognition. The parks also drive sells at the theater and stores. The Busch parks were small potatoes for Anheuser Busch and even smaller for InBev. InBev also massively slashed Anheuser Busch's ad budget.
2) Large corporations expect each business unit to pay "dividends" to the corporation, this is money that is bleed out of the business unit to pay for the corporate overlords. With a larger corporation, you could expect those "dividend" expectations to increase, removing free cash for the parks. This is a coin toss, depending on if Verizon would want to milk what is there dry, or if they wanted to invest in the future.
3) More than likely the top management would turnover. This would impact the creative direction, and type of investments. It could be good or bad, just depends.
At the end of the day, I expected Comcast to take the bleed them dry approach with Universal theme parks, but it looks like they are very committed to the parks, much more so than GE ever was. In that case though, Universal went from a unimaginably massive corporation to just a normal massive corporation. This would be the opposite direction of massive to unbelievably massive.
There is absolutely no way the parks close, and no way their names are changed away from Disney.