Stroke (specifically, a unilateral stroke affecting the brainstem or a large cerebral hemisphere): This is the most likely diagnosis because mouth deviation is a common sign of unilateral facial weakness, often seen in stroke patients. Paraparesis (weakness of the legs) can occur if the stroke affects the motor pathways in the brainstem or a significant portion of one cerebral hemisphere, leading to weakness on the opposite side of the body. Shortness of breath can be associated with stroke, especially if it involves the brainstem, which controls breathing, or if the patient has developed aspiration pneumonia as a complication.

Multiple Sclerosis (MS): MS can cause a wide range of neurological symptoms, including weakness, numbness, and vision problems. While less common, MS could potentially cause the combination of symptoms described, especially if there are lesions in the spinal cord (causing paraparesis) and brainstem (affecting cranial nerves and potentially causing shortness of breath).
Spinal Cord Injury or Lesion: A lesion or injury in the spinal cord, such as from trauma, infection, or tumor, could cause paraparesis. If the lesion is high in the cervical spine, it could also affect breathing by impairing the function of the diaphragm or other respiratory muscles. Mouth deviation might be less directly related but could occur if there's associated brainstem involvement or a separate lesion affecting cranial nerves.

Brainstem Tumor: A tumor in the brainstem could cause a combination of cranial nerve palsies (leading to mouth deviation), weakness or paralysis of the limbs (paraparesis), and respiratory difficulties (shortness of breath). This diagnosis is critical to identify early due to its potential for severe and rapid progression.
Acute Spinal Cord Compression: This is a medical emergency that requires immediate attention. Compression of the spinal cord, often due to a tumor, abscess, or hematoma, can cause rapid progression of neurological deficits, including paraparesis and respiratory failure. Early diagnosis and intervention are crucial to prevent permanent damage.

Neurosyphilis: This condition, resulting from late or tertiary syphilis infection, can cause a wide range of neurological symptoms, including stroke-like syndromes, meningitis, and dementia. While rare, it could potentially cause the combination of symptoms described, especially in the context of tabes dorsalis, which affects the spinal cord and can cause weakness and sensory deficits.
Mitochondrial Myopathies: These are a group of disorders that affect the mitochondria, often causing muscle weakness and neurological symptoms. Some forms, like Kearns-Sayre Syndrome, can affect the brain and spinal cord, potentially leading to a combination of weakness, respiratory issues, and other neurological deficits. However, the specific combination of mouth deviation, paraparesis, and shortness of breath would be unusual.