It is impossible to know whether or not the “
good thief” went “straight to heaven” or had to undergo
Purgatory. We cannot really judge that. It could be that he died in a state of perfect contrition for his sins and received baptism of desire.
A larger issue is that the Greek manuscripts that contain this passage do not contain punctuation. So did Jesus say to him: “Truly, I say to you, today you will be with me in paradise”
OR “Truly, I say to you today, you will be with me in paradise.” Punctuation makes a big difference.
Another issue is whether in this case “paradise” means heaven. It could also refer to the abode of the dead for the righteous, the bosom of Abraham. In which case the gates of heaven would not have been opened yet until the resurrection, so they would have been together in the abode of the righteous and the good thief still could have undergone some purgation before receiving the Beatific Vision. While this is a highly contested theory of the use of “paradise” it is still debated.
And, most importantly, since the cleansing of Purgatory takes place outside of space and time as we currently know them then the concept of today, tomorrow, or whenever in relation to Purgatory is not a one-to-one analogy.
Generally speaking the most common understanding of the passage has been that the good thief went “straight to heaven” due to his perfect contrition for his sins, but some other theories are also acceptable to the faith.
But I will leave you with this, Sure the word Purgatory isn't in the bible, but neither is the word "Trinity" and yet ALL Christians believe in the Triune God. Neither is the word "Incarnation" in the bible but ALL Christians believe that God became man. The word doesn't have to be there but purification of the soul is most definitely in the Bible and we call this purgatory(as in to purge):
1 Corinthians 3:11-15
"For no other foundation can any one lay than that which is laid, which is Jesus Christ. Now if any one builds on the foundation with gold, silver, precious stones, wood, hay, stubble— each man’s work will become manifest; for the Day will disclose it, because it will be revealed with fire, and the fire will test what sort of work each one has done. If the work which any man has built on the foundation survives, he will receive a reward. If any man’s work is burned up, he will suffer loss, though he himself will be saved, but only as through fire."
This passage is neither referring to Heaven or Hell on the DAY(Day of Judgement). Why? Well it's not Heaven because there is no suffering in Heaven, and it's definitely not Hell because the passage talks about still being saved and we know when you're in Hell you're there for good. So if it's neither Heaven nor Hell then what is it? It's clear from the text that it's talking about purification...
i.e ..."he himself will be saved, but only through fire"