Patients with sleep disordered breathing (SDB), also called sleep apnea, experience nighttime disrupted sleep and, because they stop breathing for short periods during the night, do not get sufficient oxygen to their brains. This can frequently result in daytime impairments including difficulties with memory. The state-of-the-art treatment for SDB is Continuous Positive Airway Pressure (CPAP). Many non-demented SDB patients who are successfully treated with CPAP show improvement in memory and other cognitive functions. Data have shown that patients with Parkinson's disease have a high rate of SDB. Patients with Parkinson's disease also often have problems with memory. This study will test the effects of treating SDB among patients with Parkinson's disease and SDB. Specifically, the study will test the effect of CPAP treatment on SDB and sleep; the effect of CPAP treatment on daytime sleepiness, cognition, overall quality of life and mood; the effect of CPAP treatment on the frequency of symptoms of REM behavior disorder and restless legs syndrome; the effect of continued CPAP use (beyond the six weeks of the study) on SDB, sleep, cognition, mood and quality of life; whether the study-partner feels that CPAP improves the patient's sleep, mood and overall functioning; whether study-partners feel that their own sleep, mood and overall functioning improve as the patient's sleep improves both during the 6-week protocol and at follow-up for those opting to continue using CPAP.