In order for JSONP to work it requires server-side support. If the API you are using does not provide any support for JSONP than you simply cannot use JSONP.
In order for JSONP to work the server must wrap the JSON it returns in a function call. Usually, an API that supports JSONP will have some sort of way of passing the name of a function it can use as a callback. This is not standard, and each API might have a different way to specify the callback. If the API doesn't support specifying a callback, than you are really out of luck.
The only other thing I can think of is if the server the API is on supports CORS. CORS is a better alternative than JSONP for making cross-domain requests, but only works on modern browsers.
The only other alternative is to have your own server proxy the request on behalf of the browser. This has the downside that all traffic to that third-party API must travel through your own application's domain.