In this paper,constrained approximate boundary controllability for linear distributed parameter dynamical systems described by partial differential equations of parabolic-type with mixed boundary conditions is considered. Using methods of spectral analysis for linear selfadjoint operators and general constrained controllability results, given in [20], necessary and sufficient conditions for constrained approximate boundary controllability are formulated and proved. Moreover, in illustrative examples constrained approximate boundary controllability of one-dimensional distributed parameter dynamical systems is investigated. The constrained controllability conditions obtained in the paper represent an extension of the unconstrained controllability results given in [7].