Do you remember the old saw, "A text without a context is a pretext"?
CONTEXT, CONTEXT, CONTEXT
Good hermeneutics requires us to contextualize a parable by asking "What came before" (and sometimes also "What came after?").
In a sense, who the "dresser" is is irrelevant. Remember, the successful interpretation of a parable depends largely on our ability to suss out the central idea of the parable--its thesis, if you will.
So, what came before the parable? Well, there was an audience to Jesus's words, and the audience comprised a variety of people, not just the "usual suspects" of scribes and Pharisees. The audience comprised a huge crowd of people, numbering in the thousands, but it also comprised Jesus's disciples. In the narrative, Jesus first addresses his disciples (Luke 12:1) and warns his inner circle to beware of the yeast of the Pharisees, which was their hypocrisy.
Then in verse 13 of Chapter 12, a person in the crowd asks Jesus to settle a personal matter. In verse 22, we are back to the audience of disciples. In verse 41, Peter, a disciple, asks Jesus a question, and in effect Peter is asking Jesus, "Are you addressing us or the crowd?" Evidently, Jesus was addressing his inner circle, and not until verse 54 does he turn his attention once again to the crowd.
Chapter 13 begins with a segment of the crowd (v.1) telling Jesus about the atrocity perpetrated by Pilate against some Galileans. Jesus responds by providing his own example of a tragedy (v.4). Together, the two events underscore the spiritual truth that sometimes bad things happen to good people. Nevertheless, Jesus insists, all people need to repent, whether they are good or evil, innocent or guilty.
To underscore the importance and necessity of repentance, Jesus tells the parable of the landowner whose fig tree failed to bear fruit. When he tells the vineyard's caretaker ("dresser") to cut the tree down, the caretaker urges restraint, at least until he gives the tree some much needed TLC.
THE MAIN POINT IS TO KEEP THE MAIN POINT THE MAIN POINT
NOW, what is the main point of the parable? That, after all, is what we must ask in order to interpret it wisely. The main point is that all people need to repent, regardless of who they are and regardless of how good or bad they are.
IN A SENSE, who the horticulturist is is beside the point. The main point of the parable is repentance.
Granted, Jesus describes God the Father as a vinedresser in John Chapter 14. There, however, Jesus is teaching clearly that the Father is in charge of pruning his true believers to make them fruitful. Here in Luke 13, however, the primary lesson is about repentance, and a secondary point is there is about a limited window of opportunity in which to repent.
In the larger context of this passage, which includes Chapter 12, those people who were killed by Pilate and those people who were killed in the collapse of a tower were dead and could not therefore repent. The people in Jesus's audience were alive and thus had an opportunity to repent, which Jesus encourages them to do.
In conclusion, the identity of the dresser in the parable is unimportant. In a sense, the fruitlessness of the tree is much more important, because lacking the fruit of repentance, the unrepentant people in Jesus's audience will certainly perish (verses 3 and 5). Food for thought.