It depends on how you define "change the key."
Traditionally, modulation is understood as a change in tonic. If you keep tonic (say, C) the same, you don't modulate. Even if it's between C major and C minor, this is not a modulation, but rather just a change in mode.
So if by "change the key" you mean "modulate," then it's only a modulation if you move to a new tonic. Otherwise, it will be a change in mode. This includes moving from, say, C Ionian to C Lydian.
Modal mixture, meanwhile, is traditionally understood as borrowing scale degrees and/or chord qualities from a parallel mode. Since parallel modes by definition have the same tonic, mode mixture thus will not qualify as a modulation.
Some commenters below seem uncomfortable with the definition of modulation I've given here. Perhaps others on the Internet use "modulation" in a different way, but music theorists are pretty consistent.
Here's Stefan Kostka and Dorothy Payne in Tonal Harmony, 5th edition, page 291:
Because parallel keys share the same tonic, we do not use the term modulation when talking about movement from one key to its parallel. The term change of mode, or mode mixture, is used instead. . . The term modulation is . . . from one tonic to another.
Miguel Roig-Francolí, in his Harmony in Context, page 540:
It should be stressed that modulation implies a change of key center. A change of mode between parallel keys, as between CM and Cm, is not considered a modulation, because the key center does not change (C in both cases).
Countless other authors, including Walter Piston, are still clear that a modulation changes the tonic pitch. In his Harmony, 5th edition, page 221:
The process involved in changing from one tonal center to another is called modulation.
It's certainly possible that others online use "modulation" to mean moving from C major to C minor. But please know that this in contrast to how modulation is traditionally taught in universities and conservatories.