Food waste is a growing global problem that originates from a variety of sources, with about 38% of it coming from food processing. In recent years, the European Union has encouraged investigations into by-products for their exploitation in several fields. In this study, the main processing by-products of artichoke (Cynara cardunculus L. var. scolymus), being leaves, stems, and external bracts, were analyzed. This study aims to valorize the by-product in order to promote its cultivation and help producers to create a new supply chain of this cultivar, typical of the Marche region in Italy, which is subject to the potential risk of genetic erosion. Several bioactive substances were monitored and quantified, including inulin, an important D-fructose polymer widely used for its physical–chemical and functional properties and prebiotic activity. Inulin extraction was optimized through an experimental design in terms of time and temperature. Moreover, the total content of polyphenols, flavonoids, and tannins was investigated in each artichoke by-product, revealing the stems as the richest fraction in all the monitored bioactive compounds.