Among primates, is it true that homosexual behavior is extremely unnatural and uncommon?
First Antonio Pardo is not a biologist, he is a professor of bioethics and his degrees are in medicine and philosophy. 

Second, Clearly he does not know anything about primates, since there is one famous great ape that has sex even more often than humans do, the Bonobo. Bonobo have sex for everything, hello, goodby, conflict resolution, social bonding, ect. Bonobo engage in homosexuality and a hundred other sexual practices, including some things humans can't do, like penis fencing. Bonobos along with chimps are also our closest living relative. 

As for other animals there are hundreds of examples of homosexuality in nature. I hate using all these wiki links but this is such general biological knowledge I would need either dozens of links or a link to a textbook. 

As for Lynn Saxon, first let me say she does not have a degree in anything, It is possible Lynn Saxon is not even her real name. 

Now for her argument, it is true that in nature sex is most often directed towards reproduction (that is why it evolved in the first place* and just consider how many sexual animals have very simple behavior), but as I have pointed out there are notable exceptions one of them our closest relative. And not just homosexuality, masturbation is observed in nature as well. Also consider monogamy is also extremely rare in nature with the exception of birds, arguing humans should behave based on how other animals act will not help their argument. It is also a common fallacy, what is natural and what is ethical are not related, Sanitation, science, religion, medicine and reading are also not natural. 

If you want to read more about humans and our relatives I suggest the book (the Third chimpanzee" by Jared Diamond as a t least a good starting book. 
"Wild Connection" by Jennifer L. Verdolin is even better, both have actual degrees in biology.

*Of course your legs originally evolved as fins,  so what something starts as is not necessarily what it is used for now.