Thin-film lithium niobate (TFLN) waveguides have emerged as a pivotal platform for on-chip spontaneous parametric down-conversion (SPDC), serving as a crucible for the generation of entangled photon pairs. The periodic poling of TFLN, while capable of generating high-efficiency SPDC, demands intricate fabrication processes that can be onerous in terms of scalability and manufacturability. In this work, we introduce a novel approach to the generation of entangled photon pairs via SPDC within TFLN waveguides, harnessing the principles of modal phase-matching (MPM). To address the challenge of efficiently exciting pump light typically in a higher-order mode, we have engineered a mode converter that couples two asymmetrically dimensioned waveguides. This converter adeptly transforms the fundamental mode into a higher-order mode, demonstrating a conversion loss of 1.55 dB at 785 nm with a 3 dB bandwidth exceeding 30 nm. Subsequently, we have showcased the device's capabilities by characterizing the pair generation rate (PGR), coincidences-to-accidentals ratio (CAR), and spectral profile of the entangled photon source. Our findings present a simplified and versatile method for the on-chip generation of entangled photon sources, which may pave the way for the application in the realms of quantum information processing and communication technologies.