The paper aims to study the mechanical properties of epoxy resin filled with clay nanoparticles (NPs), depending on their shapes and content on the surface of a modifying agent capable of forming covalent bonds with a polymer. The cylindrical clay nanoparticles with equal volume and different aspects ratios (disks, barrel, and stick) are addressed. The NPs’ bonding ratio with the polymer (RGC) is determined by the fraction of reactive groups and conversion time and varies from RGC = 0 (non-bonded nanoparticles) to RGC = 0.65 (more than half of the surface groups are linked with the polymer matrix). The performed simulations show the so-called load-bearing chains (LBCs) of chemically cross-linked monomers and modified nanoparticles to determine the mechanical properties of the simulated composites. The introduction of nanoparticles leads to the breaking of such chains, and the chemical cross-linking of NPs with the polymer matrix restores the LBCs and strengthens the composite. At small values of RGC, the largest value of the elastic modulus is found for systems filled with nanoparticles having the smallest surface area, and at high values of RGC, on the contrary, the systems containing disk-shaped particles with the largest surface area have a larger elastic modulus than the others. All calculations are performed within the framework of a mesoscopic model based on accurate mapping of the atomistic structures of the polymer matrix and nanoparticles into coarse-grained representations, which, if necessary, allow reverse data mapping and quantitative assessment of the state of the filled epoxy resin. On the other hand, the obtained data can be used to design the functional materials with specified mechanical properties based on other practically significant polymer matrices and nanofillers.