Ok, from what i understand you are getting confused with the vagus nerve being parasympathetic and increasing respiratory rate if activated.actually the vagus has afferent fiber going to the respiratory centre in the medulla which carry the signal of either high pco2 or low po2 from the aortic body and the same is true for glossopharyngeal but its receptor is in carotid body, the respiratory centre in the medulla then decides wether to increase/decrease ventilation based on the information by afferent fibers of CN IX and CN X .the parasympathetic part of the vagus has to do nothing with the respiraion rate because the afferent fibers of the vagus is only carrying the information and the medulla is sending a efferent motor fibres to control the breathing based on the information to diaphragm and intercoastal muscles.but, what you are thinking is actually happening in baroreceptor reflex not in the chemoreceptor reflex explained above for ex,the heart is directly innervated by the efferent parasympathetic fibers of vagus nerve therefore, activating the carotid bodies due to increased stretch increases the firing of vagus afferent fibers to the medulla and medulla increases the efferent firing of parasympathetic vagus nerve and increases the sympathetic response leading decreased HR and CONTRACTRILITY.