Polydimethylsiloxane (PDMS) is an important viscoelastic material that finds applications in a large number of engineering systems, particularly lab-on-chip microfluidic devices built with a flexible substrate. Channels made of PDMS, used for transporting analytes, are integral to these applications. The PDMS viscoelastic nature can induce additional hydrodynamic contributions at the soft wall/fluid interface compared to rigid walls. In this research, we investigated the pressure drop within PDMS channels bounded by rigid tubes (cellulose tubes). The bulging effect of the PDMS was limited by the rigid tubes under flowing fluids. The PDMS viscoelasticity was modulated by changing the ratio of the base to the cross-linker from 10:1 to 35:1. We observed that the pressure drop of the flowing fluids within the channel decreased with the increased loss tangent of the PDMS in the examined laminar regime [Reynolds number (Re) ∼ 23-58.6 for water and Re ∼ 0.69-8.69 for glycerol solution]. The elastic PDMS 10:1 wall channels followed the classical Hagen Poiseuille's equation, but the PDMS walls with lower cross-linker concentrations and thicker walls decreased pressure drops. The friction factor (f) for the PDMS channels with the two working fluids could be approximated as f = 47/Re. We provide a correlation between the pressure drop and PDMS viscoelasticity based on experimental findings. In the correlation, the loss tangent predominates; the larger the loss tangent, the smaller is the pressure drop. The research findings appear to be unexpected if only considering the energy dissipation of viscoelastic PDMS walls. We attributed the reduction in the pressure drop to a lubricating effect of the viscoelastic PDMS walls in the presence of the working fluids. Our results reveal the importance of the subtle diffusion of the residual oligomers and water from the bulk to the soft wall/fluid interface for the observed pressure drop in soft wall channels.