What is the most likely diagnosis for a firm, painless swelling of the lower lobe of the parotid that has grown progressively?

The diagnosis of carcinoma of the parotid is supported by the characteristics of the described swelling: firm, painless, and progressively growing. These features are often associated with malignant tumors in the salivary glands, particularly in the parotid region. Carcinomas can present as rapidly enlarging masses and may be hard in consistency due to the invasive nature of cancer cells.

In contrast, benign tumors such as pleomorphic adenoma typically present as well-defined, painless masses that grow slowly and may have a softer consistency. While these tumors can be firm as well, their growth pattern and presentation usually differ from that of malignant tumors, which can cause rapid changes in size and tenderness.

Lymphoma of the parotid may present with similar characteristics, but it is generally associated with other systemic symptoms and might be more diffuse rather than forming a discrete mass.

Benign tumors of the parotid might also match some characteristics, but their typical presentation would not usually be described as progressively growing in the same aggressive manner seen in malignant conditions. Therefore, given the noted attributes of the swelling, carcinoma of the parotid is the most plausible diagnosis among the options.