We use broad-band dielectric spectroscopy (BDS) and small-angle X-ray scattering (SAXS) to investigate the impact of architectural asymmetry in the miktoarm star copolymer on the dielectric polymer relaxations. The miktoarm copolymers studied contain one or two polystyrene (PS) chains of constant molecular weight and two identical poly(cis-1,4-isoprene) (PI) chains with varied molecular weights. Using the chains in the PI block as dielectric probes, we find that the architecturally asymmetric miktoarm star copolymer systems (PSPI2) feature distributions in chain relaxation times and dielectric relaxation strengths that are not dependent on molecular weight or morphology, in stark contrast to the effects of morphological confinement observed for symmetric diblock systems (PS-b-PI or PS2PI2). Along with evidence from the SAXS measurements regarding phase separation, these results are attributed to influences from chain stretching within the framework of the Gaussian chain model for block copolymer systems. As such, the molecular architecture in block copolymers happens to be a versatile handle to control polymer chain dynamics and, ultimately, the macroscopic physicochemical properties in architecturally complex polymers.