  For an extension $K/\mathbb{F}_q(T)$ of the rational function field over a
finite field, we introduce the notion of virtually $K$-rational Drinfeld
modules as a function field analogue of $\mathbb{Q}$-curves. Our goal in this
article is to prove that all virtually $K$-rational Drinfeld modules of rank
two with no complex multiplication are parametrized up to isogeny by
$K$-rational points of a quotient curve of the Drinfeld modular curve
$Y_0(\mathfrak{n})$ with some square-free level $\mathfrak{n}$. This is an
analogue of Elkies' well-known result on $\mathbb{Q}$-curves.
