Some examples and references are mentioned here Examples of plane algebraic curves. You can find many Jordan curves in the family $e^{it}+re^{int}, 0\leq t\leq 2\pi,$ by choosing parameters properly.

To generalize this, take any polynomial $P$ which is univalent (=injective) in the closed unit disk. The image of the unit circle is a Jordan curve. By Riemann's theorem these curves are dense in the set of all Jordan curves, so it
can be arranged that they have plenty of inflection points. A huge supply of univalent polynomials is given by the formula $P(z)=z+\epsilon Q(z)$, where $Q$ is any polynomial and $\epsilon<1/\max_{|z|=1}|Q'(z)|$. These curves depend on $\epsilon$ as parameter, and tend to the unit circle when $\epsilon\to 0$.

Another 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 $f(ke^{it}), 0\leq t\leq 2\pi d$, $d=\deg P$. Like the previous examples 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.