On an open Riemann surface R of Heins type (i.e. R is parabolic with a single ideal boundary component, \(\delta\) R), this paper considers the relationship between the harmonic dimension of the ideal boundary, dim \(\delta\) R, and the relative harmonic dimension of the ideal boundary, \(\dim_ F\delta R\). Here the former is defined as follows: For a closed parametric disk K on R, let \(HP(R-K;\partial (R-K))\) be the class of nonnegative harmonic functions on R-K whose boundary values vanish on the relative boundary \(\partial (R-K)\) of R-K. Then dim \(\delta\) R is the cardinal number of the class of nonproportional minimal functions in \(HP(R-K;\partial (R-K)).\) If K is replaced by F, where F is the union of a locally finite family of disjoint closed parametric disks on R, then \(\dim_ F\delta R\) is defined likewise. It is shown that for various choices of R and \(F: \dim_ F\delta R\dim \delta\), and \(\dim_ F\delta R=\dim \delta R\). A new proof of the known result dim \(\delta\) R\(=\aleph_ 0\) is also given.