Are there any symmetric Partial Homomorphic Encryption Schemes? Note that I am asking specifically for Partial Homomorphic Encryption (PHE) schemes and not Somewhat Homomorphic Encryption (SHE) schemes. I believe there are several SHE schemes that are symmetric but I am not familiar with any PHE schemes that are symmetric.
It depends on what you mean by "symmetric". As mentioned in the comments, you could use an asymmetric scheme and let $\mathsf{k} = (\mathsf{sk},\mathsf{pk})$. This seems unsatisfying though.

If you want symmetric to exclude this definition, a natural way to restate your question is "Are there any partially homomorphic encryption schemes in Minicrypt", in the sense that you can build them from solely the existence of a one-way function. I'm unsure of the answer in this situation, but there's some evidence against it being true.

For motivation, an easy way to construct a somewhat homomorphic encryption scheme is from $\mathsf{LWE}$.
This states that:
$$(A, As + e)\approx_c (A, u)$$
for $A$ and $e$ distributed correctly.
Standard LWE encryption schemes then use $As+e$ as a "blind", and then apply the one-time pad.
This design pattern of "Use a pseudo-random function to construct a "blind", then use the one-time pad" is fairly common. What gives LWE its homomorphic property is that the pseudo-random function $(A, As+e)$ is somewhat homomorphic.

So, a naturally way to try to answer your question would be starting with some one-way function / pseudo-random function that is partially homomorphic. A recent result showed that taking various minicrypt primitives and adding an additional partial homomorphic property is sufficient for many Cryptomania primitives (and therefore already takes you out of Minicrypt). This of course doesn't answer your question in the negative, but it does rule out a natural class of approaches to designing such a scheme while staying within Minicrypt.