A reconfigurable intelligent surface (RIS) alters the reflection of incoming signals based on the phase-shift configuration assigned to its elements. This feature can be used to improve the signal strength for user equipments (UEs), expand coverage, and enhance spectral efficiency in wideband communication systems. Having accurate channel state information (CSI) is indispensable to realize the full potential of RIS-aided wideband systems. Unfortunately, CSI is challenging to acquire due to the passive nature of the RIS elements, which cannot perform transmit/receive signal processing. Recently, a parametric maximum likelihood (ML) channel estimator has been developed and demonstrated excellent estimation accuracy. However, this estimator is designed for narrowband systems with no non-line-of-sight (NLOS) paths. In this paper, we develop a novel parametric ML channel estimator for RIS-assisted wideband systems, which can handle line-of-sight (LOS) paths in the base station (BS)-RIS and RIS-UE links as well as NLOS paths between the UE, BS, and RIS. We leverage the reduced subspace representation induced by the array geometry to suppress noise in unused dimensions, enabling accurate estimation of the NLOS paths. Our proposed algorithm demonstrates superior estimation performance for the BS-UE and RIS-UE channels, outperforming the existing ML channel estimator.