I can understand how such doctrines have evolved, but I hope fellow Christians will consider the problems with this popular argument. A key question is what is meant by "paradise"? Christ did not say "heaven." He did not tell the man that he was saved or that he would be in the presence of the Father. In fact, whatever "paradise" means, it must not be heaven, for two days later, the glorified, resurrected Christ stated that he had not yet ascended to His Father (John 20:17)
Heaven is not the same as the place called paradise. See also I Cor. 12:2-4, where Paul speaks of someone being caught up to the third heaven and of someone being caught up to paradise, as if they were different places. Paradise appears to be a place where the spirits of the dead await the time of resurrection. I don't know what Aramaic word Christ may have used, but according to my non-LDS Greek Bible Lexicon, the Greek word for paradise can mean "the part of Hades which was thought by the later Jews to be the abode of the souls of pious until the resurrection: but some understand this to be a heavenly paradise." This agrees well with what Joseph Smith said that Christ meant: "This day thou shalt be with me in the world of spirits: then I will teach you all about it and answer your inquiries" (Teachings of the Prophet Joseph Smith, p. 309). Indeed, Peter explained that when Christ was dead, he went as a spirit to preach the Gospel to those who had died (1 Peter 3:18-20; 1 Peter 4:6). Christ was not offering instant salvation to the thief, but was simply telling him that they would be in the same place that day, in the world of spirits.
The theme of Christ rescuing the souls of the dead by descending into Hades is ancient and widespread in Christianity, one that persisted into the Middle Ages but seems to have been more fully lost since the Reformation. Christ's mission of rescuing souls in hell is sometimes called the Descensus or the "Harrowing of Hell." For example, the Apostles' Creed, one of the earliest post-biblical creeds dating to the third century, affirms that Christ "descended into hell," apparently based on the teachings in 1 Peter 3:18-20. But was the place that Christ went to actually "hell"? According to the Oxford Dictionary of the Christian Church:
"Most Christian theologians believe that it [the Descensus] refers to the visit of the Lord after His death to the realm of existence, which is neither heaven nor hell in the ultimate sense, but a place or state where the souls of pre-Christian people waited for the message of the Gospel, and whither the penitent thief passed after his death on the cross (Lk. 23.43)."
[F.L. Cross and E.A. Livingstone, eds. The Oxford Dictionary of the Christian Church (Oxford: Oxford University Press, 1983), 395, as cited by Daniel C. Peterson, FARMS Review of Books, Vol. 9, No. 2, 1997, p. 137.]
There you have respected non-LDS scholars discussing the ancient Christian concept of the Descensus, the descent of Christ to rescue the dead in a place where the thief went, a place which was not heaven.
The story of the thief on the cross does teach us of the mercy of the Lord, but instead of repudiating the Lord's teachings about repentance and obedience to God, it hints at the beautiful doctrine of salvation being offered to souls in the spirit world, where both the thief and Christ would be that day, and where the thief was probably able to hear the message of the Gospel.
Related resources: Barry Bickmore's Early Christianity and Mormonism Site, and The Baptism, Descent Into Hell, and Resurrection of Unity in Christ: Early Christian Ideas and Views on Baptism by Kerry A. Shirts.