citep vs. parencite

When you are using biblatex with the natbib compatibility option (natbib=true,) to ease transition between biblatex and natbib (or vice versa),

• citep is a compatibility alias for parencite and
• citet is an alias for textcite.
• Furthermore, citealp and citealt are both aliases for cite.
• Additionally, there is some rudimentary support for natbib's defcitealias.
• Finally, there are citeauthor* and citeyearpar.

So in your example there is (and there should not be) a difference citep{knuth2001things} and parencite{knuth2001things}.

There is, however, a difference between the starred versions:

• The natbib names citep*/citet* produce citations with a local setting of maxnames=999, which means that all authors will be shown.
• The biblatex name parencite* produces citations without author names.

The natbib compatibility option will do slightly more, though. As explained in Is there a disadvantage to using natbib=true with biblatex? it will also set

DeclareDelimFormat{nameyeardelim}{addcommaspace}

---

I usually prefer not to load the natbib compatibility mode in biblatex and prefer to use the biblatex names, but muscle memory and old habits can certainly mean that the natbib compatibility mode is useful for you. Even more so if you expect to have to go back to natbib commands at some point (not many journals accept biblatex submissions).

Just keep in mind that the two quirks of natbib compatibility mode mentioned above. The behaviour of the starred version differs from standard biblatex and it sets nameyeardelim.