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