Cryptosporidium parvum is a protozoan parasite and among the most infectious diarrhea-causing pathogens, leading to severe health problems for malnourished children and immunocompromised individuals. Outbreaks are common even in developed countries, originating from water or food contamination and resulting in suffering and large costs for society. Therefore, robust, fast and highly specific detection strategies of Cryptosporidium are needed. Label-free detection techniques such as Raman spectroscopy have been suggested, however high-resolution reported spectra in the literature are limited. In this work, we report reference Raman spectra at 3 cm^{-1} resolution for viable and inactivated Cryptosporidium oocysts of the species C. parvum, gathered at a single oocyst level using a laser tweezers Raman spectroscopy system. We furthermore provide tentative Raman peak assignments for the Cryptosporidium oocysts, along with Raman mapping of the oocysts' heterogeneous internal structure. Finally, we compare the C. parvum Raman spectrum with other common enterotoxigenic pathogens: Escherichia coli, Vibrio cholerae, Bacillus cereus and Clostridium difficile. Our results show a significant difference between C. parvum Raman spectra and the other pathogens.