Parasitic infections constitute a major global public health issue. Existing screening methods that are based on manual microscopic examination often struggle to provide sufficient volumetric throughput and sensitivity to facilitate early diagnosis. Here, we demonstrate a motility-based label-free computational imaging platform to rapidly detect motile parasites in optically dense bodily fluids by utilizing the locomotion of the parasites as a specific biomarker and endogenous contrast mechanism. Based on this principle, a cost-effective and mobile instrument, which rapidly screens ~3.2 mL of fluid sample in three dimensions, was built to automatically detect and count motile microorganisms using their holographic time-lapse speckle patterns. We demonstrate the capabilities of our platform by detecting trypanosomes, which are motile protozoan parasites, with various species that cause deadly diseases affecting millions of people worldwide. Using a holographic speckle analysis algorithm combined with deep learning-based classification, we demonstrate sensitive and label-free detection of trypanosomes within spiked whole blood and artificial cerebrospinal fluid (CSF) samples, achieving a limit of detection of ten trypanosomes per mL of whole blood (~five-fold better than the current state-of-the-art parasitological method) and three trypanosomes per mL of CSF. We further demonstrate that this platform can be applied to detect other motile parasites by imaging Trichomonas vaginalis, the causative agent of trichomoniasis, which affects 275 million people worldwide. With its cost-effective, portable design and rapid screening time, this unique platform has the potential to be applied for sensitive and timely diagnosis of neglected tropical diseases caused by motile parasites and other parasitic infections in resource-limited regions.