Hi,
Pleural thickening is usually associated with asbestosis, but this sounds unlikely in you. Other possible causes include tuberculosis, sarcoidosis, rheumatoid arthritis, benign and malignant tumors, injury to the pleura.
However, it is not uncommon to have a false diagnosis of pleural thickening on xray, when it is just fat outside the pleura.
I think if you are not having symptoms it is reasonable to wait and follow up, however Xray is not very reliable for a specific diagnosis of pleural thickening. If it is still present on follow up, I would request a CT scan, and certainly if you are concerned or are having symptoms, it is reasonable to have a CT scan sooner for peace of mind.
Best of luck,
JMK MD