Sjögren's Syndrome: This autoimmune disorder is characterized by exocrine gland inflammation, particularly affecting the salivary and lacrimal glands. The patient's history of swollen parotid and submandibular glands, along with a history of seeing rheumatology, supports this diagnosis. Elevated IL-10, an anti-inflammatory cytokine, can be seen in chronic autoimmune conditions.

IgG4-Related Disease: This condition involves inflammation of various organs, including the salivary glands, and can present with swelling and hardening of the glands. The absence of other Mayo panel 13 markers does not rule out this diagnosis, as IgG4-related disease can have a varied presentation.
Sarcoidosis: This systemic granulomatous disease can affect multiple organs, including the salivary glands, and may present with gland swelling. The patient's extensive history with rheumatology and the chronic nature of the symptoms could align with sarcoidosis.

Lymphoma: Both primary salivary gland lymphoma and systemic lymphoma with salivary gland involvement are critical to consider due to their potential for severe outcomes if not promptly treated. The hard, non-remitting nature of the parotid swelling raises concern for a malignant process.
Tuberculosis or Other Granulomatous Infections: Although less common, infections like tuberculosis can cause chronic swelling of the salivary glands. Given the potential severity of these conditions, they should not be overlooked.

Heerfordt's Syndrome: A variant of sarcoidosis that specifically affects the salivary glands, often presenting with uveitis, fever, and salivary gland enlargement. It's a rare condition but should be considered in the differential due to the patient's symptoms.
Kikuchi-Fujimoto Disease: A rare, self-limiting condition that can cause lymphadenopathy and, less commonly, salivary gland swelling. It's more commonly seen in young women but can occur in anyone, making it a rare but possible diagnosis in this case.