<p>Harmonic analysis operators play a fundamental role in the study of partial differential equations, particularly in the analysis of well-posedness and regularity. In this article, we introduce a new class of weighted Bessel–Riesz operators equipped with Muckenhoupt-type weights and establish their boundedness under a variety of structural conditions. We further construct an illustrative example demonstrating that, although the classical Bessel–Riesz operator may fail to remain bounded for certain choices of exponents, its weighted counterpart continues to exhibit stable behavior. Several known boundedness results are recovered as special cases under appropriate selections of the exponents. As an application, we investigate a second-order divergence-form parabolic operator of Allen–Cahn type with coefficients in vanishing mean oscillation (VMO). Using our newly obtained estimates, we control the associated system expansion and show that the gradient of the solution belongs to Herz spaces, thereby establishing the desired regularity.</p>