Some examples and references are mentioned here https://mathoverflow.net/questions/352957/examples-of-plane-algebraic-curves. You can find many Jordan curves in the family $e^{it}+re^{ikt}, 0\leq t\leq 2\pi.$
Other examples are some certain lemniscates, they are not given as parametrized
curves ut their parametrization can be obtained. Example: $\{ z:|z^n+1|=k\}$, where
$k>1$. In general, a lemniscate is a level set of a complex polynomial $P(z)$.
It is Jordan, when the level $k$ is larger than all critical values of the polynomial.
They may have plenty of inflection points. To obtain a parametrization, you have to be able to invert the polynomial, if $f$ is the inverse, then the parametrization
is $f(ke^{it}), 0\leq t\leq 2\pi$. They have an additional advantage that they come
in continuous families, when $k$ is large, they resemble circles.

Remark. It is not a surprise that closed curves in the plane are mostly mentioned
in connection with functions complex variables. Complex variables give the most
convenient way to study them.