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:
Are there motives which do not, or should not, show up in the cohomology of any Shimura variety?