Personally, I do not believe that the Paradise which was spoken of in the Bible is the same as the Third Heaven spoken by Paul. I believe that they are two different locations and I will try my best to answer it (however anyone is free to correct me).
I believe that Paradise (which is derived from an Avestan word ‘Pairidaēza,’ meaning ‘an enclosed park or garden) is located within Sheol, also known as Hades in the Greek.
It is a common belief for the Jews, especially within the Old Testament, that when men die, they all go to a place call Sheol.
But Jacob said, “My son shall not go down with you; for his brother is dead, and he alone is left. If harm should befall him on the journey you are taking, then you will bring my gray hair down to Sheol in sorrow.”
— Genesis 42:38
But if the Lord brings about an entirely new thing and the ground opens its mouth and swallows them up with all that is theirs, and they descend alive into Sheol, then you will understand that these men have spurned the Lord.” As he finished speaking all these words, the ground that was under them split open; and the earth opened its mouth and swallowed them up, and their households, and all the men who belonged to Korah with their possessions. So they and all that belonged to them went down alive to Sheol; and the earth closed over them, and they perished from the midst of the assembly.
— Numbers 16:30-33
For there is no mention of You in death;
In Sheol who will give You thanks?
— Psalm 6:5
O Lord, You have brought up my soul from Sheol;
You have kept me alive, that I would not go down to the pit.
— Psalm 30:3
Even this idea is not foreign to the New Testament
And the sea gave up the dead which were in it, and death and Hades gave up the dead which were in them; and they were judged, every one of them according to their deeds.
— Revelation 20:13
There is an account in the New Testament which seems to fit with this understanding. In Luke 16 we see that when the Rich man and Lazarus died, both of them descended to Sheol. The rich man was suffering but Lazarus was carried to Abraham’s Bosom in Paradise.
It’s good to note now that Abraham’s Bosom does not signify a location but rather, it is a figure of speech signifying how it would be when the righteous enter into Paradise.
While commentators generally agree upon the meaning of the "Bosom of Abraham", they disagree about its origins. Up to the time of Maldonatus (AD 1583), its origin was traced back to the universal custom of parents to take up into their arms, or place upon their knees, their children when they are fatigued, or return home, and to make them rest by their side during the night (cf. 2 Samuel 12:3; 1 Kings 3:20; 17:19; Luke 11:7 sqq.), thus causing them to enjoy rest and security in the bosom of a loving parent. After the same manner was Abraham supposed to act towards his children after the fatigues and troubles of the present life, hence the metaphorical expression "to be in Abraham's Bosom" as meaning to be in repose and happiness with him.
Thus, we could also see that when the repentant thief was dying on the cross beside Jesus, he requested Jesus to remember him when He come in His Kingdom. However, Jesus assured him that today he will be with Him in Paradise.
And he was saying, “Jesus, remember me when You come in Your kingdom!” And He said to him, “Truly I say to you, today you shall be with Me in Paradise.”
— Luke 23:42-43
And indeed we know that while Jesus was in the grave, He did in fact went down to Sheol/Hades.
For Christ also died for sins once for all, the just for the unjust, so that He might bring us to God, having been put to death in the flesh, but made alive in the spirit; in which also He went and made proclamation to the spirits now in prison,
— 1 Peter 3:18-19
Therefore, we know that Christ had indeed fulfilled His statement to the Repentant thief.
Besides the Bible, we too have other sources from the early church fathers in which we can understand that after our death, we first go to Paradise in Sheol/Hades.
“For if you have fallen in with some who are called Christians, but who do not admit this[truth], and venture to blaspheme the God of Abraham, and the God of Isaac, and the God of Jacob; who say there is no resurrection of the dead, and that their souls, when they die, are taken to heaven; do not imagine that they are Christians,” - Justin Martyr (AD100-165), Dialogue with Trypho
‘That every soul is detained in safe keeping among the inhabitants of the lower world’ until the day of the Lord. The lower regions are situated as a vast, deep space in the interior of the earth. ‘By ourselves the lower regions are not supposed to be a bare cavity, nor some subterranean sewer of the world, but a vast deep space in the interior of the earth, and a concealed recess in its very bowels.’ To this region Christ went—in the manner of all dead men—so that the prophets and patriarchs might become partakers of Himself. Hence it is not unjust that the souls of the faithful should go there, too. How, indeed, shall the soul mount up to heaven, where Christ is sitting at the Father’s right hand, when as yet the archangel’s trumpet has not been heard by the command of God, when as yet those whom the coming of the Lord is to find on the earth have not been caught up into the air to meet Him at His coming, in company with the dead in Christ, who shall be the first to arise? ‘When the world indeed shall pass away, then the kingdom of heaven shall be opened.’12
The souls of all shall pass into the ‘lower world,’ where they shall remain until the resurrection. Good and bad alike are there. But not together, in the same place. -Tertullian (AD 160), De Paradiso
Therefore once more, we can see that Paradise is situated within Sheol/Hades where the righteous abide until the Day of the Resurrection.
Thus far I can conclude that Paradise is indeed a different place in comparison with the Third Heaven. As for the Third Heaven, I will say that there is where the Throne of God is located. The very place which John the Apostle saw in his vision in Revelation.
After these things I looked, and behold, a door standing open in heaven, and the first voice which I had heard, like the sound of a trumpet speaking with me, said, “Come up here, and I will show you what must take place after these things.” Immediately I was in the Spirit; and behold, a throne was standing in heaven, and One sitting on the throne.
— Revelation 4:1-2
Besides that, I hope that I have covered enough to explain that the Third Heaven is not the same as Paradise.
