This paper investigates the vertical response of end-bearing piles in a homogeneous isotropic linear elastic soil on a rigid bedrock subjected to a vertical harmonic point load at the soil’s surface. A numerical model is used to compute vibration responses. A novel system of dimensionless parameters is established to bring insight into the influence of the relationships between soil and pile properties on the dynamic pile–soil interaction and to allow for general conclusions to be drawn. The results show the conditions under which the relationship between the axial stiffness of the pile and the stiffness of the soil has a significant influence on the end-bearing pile response. Different pile group configurations are considered where the vertical response is found to be bounded by the single pile response, justifying its use as a conservative estimate for the group response. Finally, an expression including only two dimensionless parameters, the pile slenderness ratio and the pile–soil stiffness ratio, is proposed for calculating an estimation of the vertical response of an end-bearing pile from the free field vibrations.