There is no problem with constructing an abelian variety $A$ for _most_ Hilbert modular forms of parallel weight $2$, the issue is finding such a variety for _all_ $\pi$. In particular, when $d = [K:\mathbf{Q}]$ is even, there is a local obstruction to the existence of a corresponding Shimura curve which realizes the Galois representation associated to $\pi$.
In particular, if $\pi$ has "level one", then no such Shimura curve exists.
 To construct the Galois representation in this case one has to use congruences; this was done by Taylor in the late 80's. 
This issue is also discussed here:

http://mathoverflow.net/questions/39485/are-there-motives-which-do-not-or-should-not-show-up-in-the-cohomology-of-any-s