It is shown that, for every noncompact parabolic Riemannian manifold $$X$$ and every nonpolar compact $$K$$ in $$X$$ , there exists a positive harmonic function on $$X\setminus K$$ which tends to $$\infty $$ at infinity. (This is trivial for $$\mathbb{R }$$ , easy for $$\mathbb{R }^2$$ , and known for parabolic Riemann surfaces.) In fact, the statement is proven, more generally, for any noncompact connected Brelot harmonic space $$X$$ , where constants are the only positive superharmonic functions and, for every nonpolar compact set $$K$$ , there is a symmetric (positive) Green function for $$X\setminus K$$ . This includes the case of parabolic Riemannian manifolds. Without symmetry, however, the statement may fail. This is shown by an example, where the underlying space is a graph (the union of the parallel half-lines $$\left[0,\infty \right)\times \{0\}, \left[0,\infty \right)\times \{1\}$$ , and the line segments $$\{n\}\times [0,1], n=0,1,2,\dots $$ ).