Check your style defintitions. Dollars to donuts Times is in [Basic Paragraph] or some other style in the receiving document. Any styles that are named the same will use the destination doc definitions.
As I said, the default font is Minion Regular. The font set in the default Basic Paragraph style is Minion Regular.
The reason Peter would ask that what you are reporting sounds exactly like a situation where your source document from which you are copying has a different Basic Paragraph style from the document into which you are pasting. So, if you only looked at the Basic Paragraph style in your source doc, it wouldn't tell you anything. I'm perfectly willing to assume that you checked both source and target style definitions, I guess.
However, there are a few other ways to generate similar results - e.g. let's say you have frames defined in your source doc with the Basic Text Frame Object Style - note that this is an object style, not a text style. However, it's possible to specify a paragraph style in an object style, and maybe the Basic Text Frame in your target doc specifies a paragraph style with Times? Something similar can be done with Table Styles and Cell Styles, are there any tables in what you are copying and pasting?
These are, obviously, just guesses. Other guesses (without any evidence to support 'em) might be document corruption (export INX or IDML, re-open and re-save) or remotely possible preferences corruption or application corruption (reset preferences, or reinstall app).
Lastly, if you can (you may be working on something sensitive, or covered with Do Not Distribute flags), you may just want to post a link from which the forum regulars can download your misbehaving InDesign files. Sometimes, what takes dozens of careful messages to uncover can be found in thirty seconds of clicking.