This may answer your question, but its more of a broad cultural sense than Romans specifically thanking their veterans. I will be talking mostly of the Early Principate, though most of this applies to the Late Republic as well.

The military of Rome was seen as the heart of Roman-ness, and for good reason. It was the normal way to get into politics (as being a good administrator and general were seen as the same), one of the few ways to gain citizenship, one of the few ways to gain land, and was in general regarded as the most Roman profession, next to farmer. As well, it was regarded as the best way to teach Roman values, again next to farmer. These ideas can be seen very clearly in On Agriculture and many of the Late Republican speeches. I will get some exact passages when I have some copies in front of me.

Now, it should also be noted that Romans culturally and in literature loved the idea of a valorous soldier distinguishing himself in battle. That's how you got promotions in the Principate and land in the Republic. There's so many points where this can be described in literature, from pretty much all of Livy's From The Founding Of Rome to Caesar's Conquest of Gual. We have plenty if names from soldiers who weren't important politicians, prolific writers, or "Great Men". They were just soldiers who distinguished themselves and were either given splendid altars by relatives or written in the literature of their Generals (looking at you, Caesar).

Finally, I'd also like to note that a veterans service never really ended. When they were discharged, they got a plot of land to farm on in the province they were assigned to and some citizenship. However, they were often called up back to service to defend their province. A notable case of this is during Bouddica's rebellion. The Romans were able to double their numbers (from 5,000 to 10,000) by sending messages to any former soldier who could still hold a gladius.

Im not sure if this qualifies as your definition of "thanking veterans" as you wanted, but obviously military service was rewarded, both physically (land, citizenship) and in more abstract forms (veneration of the legionnaire)