Stroke (Middle Cerebral Artery Infarction): This is the most likely diagnosis because a stroke, particularly in the middle cerebral artery territory, can cause contralateral hemiparesis (weakness on the opposite side of the body) and ipsilateral conjugate gaze palsy (difficulty moving the eyes towards the side of the lesion). The middle cerebral artery supplies a significant portion of the lateral surface of the brain, including areas responsible for motor function and eye movement.

Brainstem Stroke: Although less common than a hemispheric stroke, a brainstem stroke can also cause gaze palsy and hemiparesis, depending on the specific location and structures involved.
Traumatic Brain Injury: Trauma to the brain can result in a variety of focal neurological deficits, including gaze palsy and hemiparesis, depending on the areas of the brain affected.
Intracranial Tumors: Tumors in the brain, such as gliomas or metastases, can cause localized damage leading to gaze palsy and hemiparesis by compressing or infiltrating brain tissue.

Cerebral Venous Thrombosis: This condition, although less common, can present with a variety of neurological symptoms, including focal deficits like gaze palsy and hemiparesis. It is crucial to consider this diagnosis because anticoagulation is the treatment of choice, which differs significantly from the treatment for other conditions like ischemic stroke.
Intracranial Aneurysm or Arteriovenous Malformation (AVM): These vascular abnormalities can cause focal neurological deficits by compressing adjacent brain tissue or by rupturing and causing a hemorrhagic stroke. Early detection is critical for preventing further complications.

Multiple Sclerosis: While multiple sclerosis (MS) can cause a wide range of neurological symptoms, the specific combination of conjugated gaze palsy and hemiparesis would be unusual as a presenting feature. MS typically presents with more disseminated symptoms in time and space.
Progressive Supranuclear Palsy (PSP): This is a rare neurodegenerative disorder that can cause supranuclear gaze palsy, among other symptoms. However, the gaze palsy in PSP typically involves vertical gaze and is progressive, which might distinguish it from the acute onset of conjugated horizontal gaze palsy.
Encephalitis: Inflammation of the brain due to infection can cause focal neurological deficits, including gaze palsy and hemiparesis, but this would typically be accompanied by other symptoms such as fever, altered mental status, and seizures.