Nontrival primes: human chromosome number is 23, Sunflower chromosome number is 17, both of which are prime. Most flowers have an odd and often prime number of petals: five is a common number.
I assume you really meant non-trivial primes, where $p>2$ or $p>3$, but you didn't specify that, so let me point out some basic regions where 2 and 3 are prominent. And $2$ and $3$ are very prominent through-out nature, so these are not very special concepts.
Diploid genomes come in copies of 2, and 2 is prime. DNA chains come in duplicate copies, with one side reading in one direction as the sense and the other side being the "inverted carbon copy" and called the nonsense side. When DNA chains are copied, they split apart like a zipper and complementary copies are made on both, yielding two identical doubled-DNA chains.
Three is also a prime number:
The human chromosomal set consists of 23 pairs of DNA chromosomes: $46$ in total, with $44$ of those being non-sex chromosomes, and $2$ of them being sex chromosomes: $XX$ for females and $XY$ for males.
Also, when you convert from 2 copies of these chromosomes to three, for example Trisomy 21, you can end up with Down's syndrome.
If, instead of having two sex chromosomes of the usual type, $XX$ or $XY$, you can have XYY syndrome or XXY syndrome also known as Klinefelter's syndrome. Technically, you could say that these types of sex-chromosome sets are not in the set of the usual two genders of male ($XY$) and female ($XX$), but are actually outside of the two genders. So to answer @vonjd's comment to the question, there are not just 2 genders.
Extra random point: the sunflower's spiral follows Fermat's spiral which at various points, is a prime number. So there must be some sunflowers which have a prime number of sunflower seeds.
Many flowers have five petals, five is prime.