A testing paradigm is a mental framework of testing. It covers the way you think about testing, the way you approach deciding your testing methods, the methodologies you choose to use, and even the words you use to describe the things you are doing.
For example, the Context-Driven school of testing follows a paradigm that says, more or less, that the context in which you are working is the single most important influence on what you do and how you do it. The Agile Manifesto states the same thing, but by implication rather than explicitly. The Waterfall methodology follows a different paradigm, one that states that it is possible to know enough about a software development project to effectively map everything out from the start and follow the plan.
A testing method is just a way to approach testing. Black-box is a method: it is approaching testing as though you know nothing about the internal workings of the software so the software is a "black box" where specific inputs and actions are expected to produce specific outputs.
The same testing method can be used in different testing paradigms: black-box testing is used with multiple test methodologies, which in turn belong to different testing paradigms.