The definitions of a parameter and statistic pretty much agree: parameters and statistics are numerical characteristics or numerical summaries of a population and sample, respectively, for a given study. I don't think this is common usage, but…
Could the population size $N$ be considered a parameter? Could the sample size $n$ be considered a statistic?
After all, the size of the population or sample is a numerical summary or characteristic of the population or sample.
A “statistic” has the rather trivial definition of being a function of the data, so counting how many points there are is a function of the data. Sure, sample size is a statistic.
A “parameter” is a knob you turn to get some distribution to behave a certain way. If you want a normal distribution centered at 7, turn $mu$ up to 7. If you want it spread out a lot, turn $sigma^2$ up to 81.
“Population size” is a strange idea, and you can find differing opinions about if it can exist. You may think that if you observed every person, then you’ve observed the population. Say you found that people now are taller than people 200 years ago, having measured everyone in 1819 and 2019. If you then do a hypothesis test of their heights, you’re saying that what you’re interested in is the process that generates human heights, and the humans you observed are the ones who happened to be born.
I say that a parameter is some characteristic of a distribution (in the mathematical sense of being a CDF). Therefore, population size is $infty$ and not a parameter.