The "Oliver Twist" part is pretty obvious, so we won’t belabor that (see the "Character Analysis" and "Quotes" sections for more on Oliver’s name). But the subtitle could probably use more explanation.
What is a "parish boy," anyway? Well, in nineteenth-century England, taking care of poor people was the responsibility of the national Church of England (a.k.a. the Anglican Church, or, as they call it in the US nowadays, the Episcopal Church). And the Church of England was divided into local communities, or "parishes."
If you were poor, unable to work, or an orphan, the parish in which you were born took care of you. (Of course, it didn’t always work out that way…) So, if you were an orphan being looked after by your local parish, you were referred to (disparagingly) as a "parish boy" (or girl). The sub-title of Oliver Twist puts Oliver’s story in a broader social context. It also juxtaposes Oliver’s own, particular (and fictional) life story with something more universal—it could be the story of any "parish boy." And there were an awful lot of them.
Finally, the "progress" part: a progress is a particular kind of narrative, that (surprise, surprise) tells the story of a person’s progress through the events of their life. This kind of story tends to be pretty episodic (easily broken down into recognizable events or episodes). One famous example is John Bunyan’s The Pilgrim’s Progress (1678), which is an allegory about the progress of a Christian guy creatively named "Christian" as he moves through various life episodes called things like "the Valley of Humiliation," "the Slough of Despond," and "Vanity Fair" (yes, that’s where the magazine name came from!).
Another famous "progress" narrative is the set of pictures by the artist William Hogarth called "A Rake’s Progress" and "A Harlot’s Progress" (1730-1735), showing how country boys and girls can fall in with a bad crowd and turn into criminals, gamblers, and prostitutes in the big bad city. The Hogarth connection is particularly important because Oliver Twist is all about how being turned loose in London can expose you to bad influences, and might turn you into a criminal. Both the Christian allegory by Bunyan and the Hogarth pictures were wildly popular in Dickens’s day, so the parallels would have been immediately obvious to his contemporary readers.