Identifying the specific red color you saw would require knowing the species of mushroom. Many produce a wide range of such pigments and has been the source of some investigation. See for example
Current Opinion in BiotechnologyVolume 16, Issue 2, April 2005, Pages 231-238 Plant biotechnology/Food biotechnology
Exploring fungal biodiversity for the production of water-soluble pigments as potential natural food colorants
Sameer AS Mapari1, Kristian F Nielsen1, Thomas O Larsen1, Jens C Frisvad1, Anne S Meyer2 and Ulf Thrane1,