This paper deals with the motion planning problem for parallel orienting platforms with one non-holonomic joint and two prismatic actuators which can maneuver to reach any three-degree-of-freedom pose of the moving platform. Since any system with two inputs and up to four generalized coordinates can always be transformed into chained form, this path planning problem can be solved using well-established procedures. Nevertheless, the use of these procedures requires a good understanding of Lie algebraic methods whose technicalities have proven a challenge to many practitioners who are not familiar with them. As an alternative, we show how by (a) properly locating the actuators, and (b) representing the platform orientation using Euler parameters, the studied path planning problem admits a closed-form solution whose derivation requires no other tools than ordinary linear algebra.