I don’t want to discredit Dottard’s answer. It is basically what I was taught as a child. Neither do I consider it the wrong answer, but give it +1. But, I want to consider all the available information.
Much information I will get from Bailey, K. E. (2008), Jesus through Middle Eastern Eyes: Cultural Studies in the Gospels, Downers Grove, IL: IVP Academic. He was a professor in a seminary in Beirut, Lebanon, from 1967 to 1984.
Many have pointed out that this is the only parable where someone is named, Lazarus. Some even suggest, because of the proper name, this is an actual account. In either case apparently Jesus intended it to be interpreted as a parable teaching one central true rather than an allegory. Thus, the details are only relevant to supporting that point. Thus, the details about the dogs are relevant when characterizing the rich man and Lazarus’s condition portrayed at the beginning of the parable.
Lazarus was ἐβέβλητο (pluperfect passive indicative of βάλλω) literally thrown there and left.
Covered with sores, Lazarus was too weak to work or sit up, and he “desired to be fed” with what “fell from the rich man’s table.” The verb desired [ἐπιθυμῶν] is used in Luke’s Gospel for something a person wants but is unable to have. … As reference is made to leftovers from meals, a Middle Easterner immediately thinks of the estate’s guard dogs, who are the natural recipients of the remnants of any meal. -- Bailey, K. E., p. 384
Were the dogs adding to Lazarus’s suffering or in contrast to that allowed by the rich man?
In reference to these dogs the text reads, “But [alla] the dogs came and licked his sores.” The Greek word alla always indicates a contrast. Invariably, the English language tradition has understood the dogs’ actions to be in harmony with the cruelty of the rich man, … But for more than a thousand years most Arabic versions have accurately translated the Greek word alla as a contrast, and thereby emphasized that the dogs were not joining the rich man in tormenting Lazarus. This contrast is clear in the Greek text and is important to the story. -- Bailey, K. E., p. 385.
Were the dogs licking an act of contempt or compassion?
Dogs lick their own wounds. They lick people as a sign of affection. But more than this, recent scientific scholarship has identified that saliva contains “endogenous peptide antibiotics,” which facilitate healing. A dog’s saliva contains such peptide antibiotics, and the ancients somehow discovered that if a dog licked wounds they would heal more rapidly. -- Bailey, K. E., p. 385.
The following discovery indicates an awareness of the healing from dogs licking wounds.
In 1994 Professor Lawrence Stager of Harvard University discovered more than 1,300 dogs buried in ancient Ashkelon. The graves date from the fifth to the third centuries B.C. when Ashkelon was ruled by the Phoenicians. These animals were probably linked to a Phoenician healing cult. -- Bailey, K. E., p. 385.
Guard dogs or strays?
Dogs in the Middle East are not pets. Elsewhere in Scripture they are always seen in a negative light (Is 56:10; 66:3; Phil 3:2; Rev 22:15) and often mentioned in connection with pigs (Is 66:3; 2 Pet 2:22). In the early Jewish tradition dogs were considered almost as unclean as pigs. The Mishnah notes, “None may rear swine anywhere. A man may not rear a dog unless it is kept bound by a chain.” Such dogs were kept as guard dogs. The dogs in this story may be wild street scavengers, but in that case the rich man’s servants would have driven them away for the benefit of his guests. -- Bailey, K. E., p. 386.
Ibn al-Tayyib, the monk, biblical scholar and medical doctor, comments:
I understand that the licking of Lazarus’s sores gave him relief and eased his pain. This reminds us that the silent, unspeaking animals felt compassion for him and they helped him and cared for him more than the humans. He was naked without medical attention other than what he received from the dogs. This demonstrates that the rich man did not notice him or give him any attention at all. Thus when we compare the rich man’s condition to that of Lazarus, we see that the first was clothed with purple and linen. The second was naked and covered with sores. The first luxuriated every day with a banquet while the second longed for scraps of bread. The first had many servants ready to satisfy all of his needs and the other had no servants other than the dogs. (Ibn al-Tayyib, Tafsir al-Mashriqi, 2:292-93)
-- Bailey, K. E., p. 386.
It is unlikely that Lazarus was at the gate of the rich man if he never got any food. The passage does not tell us how Lazarus got food. It is unlikely that they threw out scraps to feed stray dogs. That would only attract them. Guard dogs would have been in the yard on the other side of the gate. The scraps would be given to them. Thus, it is quite possible that the dogs had enough sympathy with Lazarus to bring him some of the scraps.
Guard dogs weren’t friendly to strangers. Neither would stray dogs be in a culture that did not have dogs as pets and chased off stray dogs. The dogs licking Lazarus’ wounds said something about Lazarus, that he was a kind and friendly person to the dogs, so they were friendly to him.
The rich man’s actions valued Lazarus no better than a dog, but the actions of the dogs put the rich man’s morality at a lower level than the dogs.