I did not mean to imply that Wikipedia was my source for the etymology of even and odd as classifications of functions. (I hadn't really meant to make an etymological point at all, and I added the link only so that readers unfamiliar with the concept of Taylor series might be enticed to click through and learn something.)

I don't have any references for you right now, but I would be surprised if the usage of the terms doesn't pre-date your citation of Euler by at least several decades. The basic technique for deriving Taylor series is known to date to at least 1671, to the correspondence between James Gregory and John Collins; it also shows up in letters from de Moivre to Johann Bernoulli in 1694 and from Leibniz to Bernoulli in 1708. Newton had a geometric means of deriving the coefficients of power series when he was writing the Principa (published in 1687), as demonstrated by the proof of his tenth proposition, and he included a description of the algebraic technique in an early draft of his Quadrature of Curves (but removed it before publication in 1706). So Taylor series were broadly known to Europe's prominent corresponding mathematicians in the first decades of the eighteenth century, and to some of them decades earlier.

And the basic concept of power series, as well as the power-series representations of many common functions, were already widely known in the latter decades of the seventeenth century. Like I said, I don't have a reference for you at the moment, but I have always heard that the terms odd and even derive from the power-series representations of functions, and it's difficult to imagine that no one before Euler had noted that the power series of some functions involve only odd or only even powers—although one may more readily imagine that someone noted it while neglecting to name it.

Anyway, Taylor series and the more basic concept of power series certainly were known to Euler when he wrote the paper you cited. Since the power series of polynomial functions are the polynomials themselves, you're right that Euler would not have had power series in mind in the context of this paper. (But, no, Euler would not have been thinking in terms of smoothness—mathematicians up until the analytic enlightenment of nineteenth century played fairly fast and loose with derivatives. The point you made about ratios of functions is off-base, I think: the paper is about reciprocal solutions to polynominal equations.)

The question is simply whether your citation is evidence that Euler first coined the terms for the concept in the context of this paper (i.e., at that time and referring specifically to polynomials), or whether the concept and terminology already existed. If not, when was it generalized to refer to functions other than polynomials of finite order? I don't know. Perhaps nobody bothered to give the concept a name until Euler in 1727, and perhaps nobody even considered the concept with regard to a more general class of functions until even later. (I have a source book around here somewhere that might say something about the matter. ...)

In any event, if you asked a sample of mathematicians today why functions are called odd or even, I'd bet that many of them would point to power-series representations. That's what I meant to point out in my earlier comment.