<p>We elaborate on the method of parametric annihilators for deriving integral relations. Parametric annihilators are differential operators that annihilate multivalued integration kernels appearing in suitable integral representations of special functions. We illustrate this approach in a way that applies to a broad variety of integral representations. We describe a method for computing parametric annihilators based on efficient linear solvers and use them to derive relations between a wide class of special functions related to important problems in high-energy physics. We also formulate a similar method for deriving differential equations satisfied by the independent integrals within an integral family. We show applications to several classes of special functions, including hypergeometric functions, loop integrals in various representations (including Baikov, loop-by-loop Baikov, Lee-Pomeransky and Schwinger representations) and duals of loop integrals. We finally present the public M<span>athematica</span> package CALICO for computing parametric annihilators and its usage in several examples of high relevance in theoretical particle physics.</p>