Industrial almond by-products (ABPs, almond oil cake) and tomato by-products (TBPs, mixture of peel and seeds) are nutrient-rich residues that could enhance the nutritional value of gluten-free pasta and support sustainable waste reduction. This study investigated the impacts of ABP or TBP incorporation and the extrusion screw speed on rice-based pasta quality. Rice pasta (control) and pasta enriched with ABP or TBP at different rates (2.5%, 5.0%, 7.5% and 10.0%) were processed by extrusion-cooking at two screw speeds (50 and 90 rpm), and their quality (cooking loss, water absorption capacity, hardness, firmness, adhesiveness and color) was evaluated, as well as energy requirements during processing (specific mechanical energy (SME)). The results showed that an increase in the ABP and TBP rates had a more pronounced positive effect on cooking loss, water absorption capacity, yellowness and redness values. Moreover, comparable hardness and firmness, but lower adhesiveness values, were recorded in cooked rice–ABP-enriched pasta. However, cooked pasta enriched with TBP had lower hardness than the control pasta. Regarding energy consumption, pasta processed with a higher by-product rate at a higher screw speed showed similar SME as compared to the control pasta. In this regard, ABP can be used successfully from 7.5 to 10.0% in rice–ABP pasta formulations without compromising its technological properties. However, the optimization of the additive rate of TBP requires further study.