The word 'today" stands between the two clauses and Greek usage would permit the writer to put it anywhere in the sentence. Also there was no punctuation in the original texts so Jesus could have been saying "I say unto to you this day" that you will be with me in paradise.
Also the usage of paradise would have a connotation of a park or preserve or some kind of enclosed area. The Garden of Eden would be an example.
Lastly let us remember the teachings of Jesus on difficult verses in this case specifically of His promise to "reward every man according to his works" when? when He returns in triumph "in the Glory of His Father with his angels". There are many more supportive verses to add clarity on this subject.
“There are many more supportive verses to add clarity on this subject.”
True, and I think that is the most important point. There are a great many verses supporting the view that final rewards or final condemnations do not come until the final judgement, after the second coming of Christ. However, the people supporting the opposite view only seem to ever be able to cite this one isolated verse, whose meaning is ambiguous if you study it critically.