Mucoepidermoid Carcinoma with High-Grade Features or Adenocarcinoma: This diagnosis is most likely due to the presence of skin ulceration and facial nerve weakness, which are indicative of aggressive tumor behavior. Mucoepidermoid carcinoma is one of the most common malignant salivary gland tumors and can present with a range of aggressiveness.

Squamous Cell Carcinoma: Given the involvement of the skin with ulceration, squamous cell carcinoma, either primary in the parotid gland or metastatic, is a consideration. It is known for its aggressive nature and potential to cause facial nerve weakness if it invades the nerve.
Salivary Duct Carcinoma: This is a highly aggressive tumor that can present with rapid growth, skin invasion, and facial nerve paralysis, making it a plausible diagnosis in this scenario.
Carcinoma Ex Pleomorphic Adenoma: Although less common, this malignant transformation of a pleomorphic adenoma can exhibit aggressive behavior, including facial nerve involvement and skin ulceration, especially if the malignant component is high-grade.

Metastatic Disease to the Parotid Gland: It is crucial not to miss the possibility of metastasis from another primary site, such as the skin (e.g., squamous cell carcinoma) or other head and neck cancers. The presence of skin ulceration and facial nerve weakness could be indicative of an aggressive metastatic disease.
Lymphoma Involving the Parotid Gland: Although less likely to cause skin ulceration, lymphoma can involve the parotid gland and, in rare cases, cause facial nerve weakness. It's a diagnosis that should not be overlooked due to its different treatment approach.

Acinic Cell Carcinoma with High-Grade Transformation: While acinic cell carcinoma is generally considered to have a better prognosis, high-grade transformation or dedifferentiation can lead to aggressive behavior, including facial nerve invasion and skin ulceration.
Primary Parotid Lymphoepithelial Carcinoma: A rare and aggressive tumor that can present with similar symptoms, including facial nerve weakness and skin involvement, although it is more commonly associated with Epstein-Barr virus in the Asian population.
Other Rare Salivary Gland Malignancies (e.g., Myoepithelial Carcinoma, Hyalinizing Clear Cell Carcinoma): These tumors are rare and can exhibit a range of behaviors, but their consideration is important in the differential diagnosis of a parotid mass with aggressive features.