Some examples and references are mentioned here Examples of plane algebraic curves. You can find many Jordan curves in the family $e^{it}+re^{iat}, 0\leq t\leq 2\pi,$ by choosing parameters properly.
A large class of examples are certain lemniscates, they are not given as parametrized curves but their parametrization sometimes 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. (For other $k$ not equal to moduli of critical values, they are disconnected unions of Jordan curves). 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 $f(ke^{it}), 0\leq t\leq 2\pi d$, $d=\deg P$. They have an additional advantage that they come in continuous families; when $k$ is large, they resemble circles. A theorem of Hilbert says that every Jordan curve can be approximated by lemniscates so the variety of their shapes is unlimited, but of course most of them cannot be explicitly parametrized. Still those which can be give plenty of examples.
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 describe and study them.