This paper addresses the dynamic analysis of nearly periodic structures composed of substructures (cells) with varying properties. This may concern 2D or 3D substructures subjected to geometric modifications (mesh variations) or more standard parametric changes. In parametric model order reduction, reduced matrices of substructures can be interpolated over a multi-dimensional parametric space. Also, an interface reduction between the substructures can be proposed in which the vectors of interface degrees of freedom are described using the interface modes of an equivalent purely periodic structure. To improve the accuracy of the interpolation strategy at high frequencies, basis enrichment techniques are proposed. Within this framework, high-order static modes are used to enrich the basis of component modes of the substructures. Also, static correction vectors are used to enrich the basis of interface modes to account for the varying properties of the substructures.