It's not Paradise. That's usually understood. This is the "place" in the afterlife otherwise called Abraham's Bosom, and its role has been discussed here before. So, the question isn't whether or not such a place existed but how Christ's reference to Paradise makes sense when he makes his famous reply to the thief.
While the popular wisdom is that Christ is assuring him, in response to his somewhat guarded profession of faith in Christ, that he will be with Christ in heaven that very day, Paradise is what is mentioned, not heaven, so another explanation is in order. This is that Christ is saying that the two of them will be done with their suffering that very day and will both pass into a better situation in the (non-physical) afterlife. That's entirely in accord with the exchange between the two of them.
Let me get my head around this. The word used is "Paradise" but you say it's not. Can you prove that? The Greek word used here - paradisos - is the same one used in Revelation 2:7 - "the paradise of God". Just because it's been discussed here doesn't lend weight to this line of view.
"Paradise is what is mentioned, not heaven". So what's the "paradise" in Revelation 2:7?
Can you just face that there are multiple problems here? The thief doesn't repent according to Matthew and Mark. The usage of "Paradise" is pretty sparse, most people understand it to be heaven, and that's how it's used in Revelation 2:7. The other big problem is this, according to Revelation 20, the dead aren't raised until the Judgement. Also John 5:29. In other words, the dead have to wait. Until Judgement day.
But if we DO NOT make Jesus say that he is indeed going to come into his kingdom that very day, etc., it makes sense. Promising salvation or heaven to the thief isn't actually a part of what Jesus said there, even if it is part of what was said to him by the thief.
It still doesn't make sense in light of Matthew and Mark's crucifixion accounts. They make no mention of any repentant thief. One would think this would be sort of important because it got a response from Christ...but nope, it's totally absent.
Again, in Luke's version the thief asks to be remembered when Christ comes into his Kingdom. That certainly couldn't be that day, or even the 3rd day. It has to be on Judgement day, because that is what is taught, and "Paradise" means paradise (heaven/kingdom of God etc) according to the regular usage of the word in the Greek.