Parshurama did not refuse Karna because he was a Suta but because a Suta is part Kshatriya. Manu Smriti Chapter X mentions who a Suta is:
- From a Kshatriya by the daughter of a Brahmana is born (a son called) according to his caste (gati) a Suta; from a Vaisya by females of the royal and the Brahmana (castes) spring a Magadha and a Vaideha.
Parshuram's enmity was with the Kshatriyas and therefore he took students only from non-kshatriya Varnas. He cursed Karna because he had lied to him about his real Varna and the entire episode is mentioned in SECTION III of Shanti Parva:
Then Rama wrathfully addressed Karna, saying, 'O fool, no Brahmana could endure such agony. Thy patience is like that of a Kshatriya. Tell me the truth, without fear.' Thus asked, Karna, fearing to be cursed, and seeking to gratify him, said these words, 'O thou of Bhrigu's race, know me for a Suta, a race that has sprung from the intermixture of Brahmanas with Kshatriyas. People call me Karna the son of Radha. O thou of Bhrigu's race, be gratified with my poor self that has acted from the desire of obtaining weapons. There is no doubt in this that a reverend preceptor in the Vedas and other branches of knowledge is one's father. It was for this that I introduced myself to thee as a person of thy own race.' Unto the cheerless and trembling Karna, prostrated with joined hands upon earth, that foremost one of Bhrigu's race, smiling though filled with wrath, answered, 'Since thou hast, from avarice of weapons, behaved here with falsehood, therefore, O wretch, this Brahma weapon shalt not dwell in thy remembrance. Since thou art not a Brahmana, truly this Brahma weapon shall not, up to the time of thy death, dwell in thee when thou shalt be engaged with a warrior equal to thyself! Go hence, this is no place for a person of such false behaviour as thou! On earth, no Kshatriya will be thy equal in battle.' Thus addressed by Rama, Karna came away, having duty taken his leave. Arriving then before Duryodhana, he informed him, saying, 'I have mastered all weapons!'"
The reason for Parshurama's enmity with Kshatriyas is mentioned in Section CXVII of the Vana Parva:
And Rama, the conqueror of hostile cities, cremated his father on the funeral pyre, and vowed, O scion of Bharata's race, the slaughter of the entire military caste, and of exceeding strength in the field of battle, and possessed of valour suited to a heroic soul, and comparable to the god of death himself, he took up his weapon in wrathful mood, and singlehanded put Kartavirya's sons to death. And, O chieftain of the military caste,** Rama, the leader of all capable of beating their foes, thrice smote down all the Kashatriya followers of Kartavirya's sons. And seven times did that powerful lord exterminate the military tribes of the earth.** In the tract of land, called Samantapanchaka five lakes of blood were made by him.
As regards the Pandava's (mis)treatment of Karna, it can be understood from the point of view of Karna's chalenge to Arjun on the day of their graduation from the Gurukul of Dronocharya and his subsequent friendship with Duryodhan.
Section CXXXVIII of the Sambhava Parva can help us understand this better:
"Vaisampayana continued, 'When the spectators, with eyes expanded with wonder, made way for that subjugator of hostile cities, Karna, that hero with his natural mail and face brightened with ear-rings, took up his bow and girded on his sword, and then entered the spacious lists, like a walking cliff [......] And that foremost of eloquent men, the offspring of the Sun, in a voice deep as that of the clouds, addressed his unknown brother, the son of the subduer of the Asura, Paka (Indra), saying, 'O Partha, I shall perform feats before this gazing multitude; excelling all thou hast performed! Beholding them, thou shall be amazed.'
And, O thou best of those blest with speech, he had hardly done when the spectators stood up all at once, uplifted by some instrument, as it were. And, O tiger among men, Duryodhana was filled with delight, while Vibhatsu (Arjun) was instantly all abashment and anger.
Then with the permission of Drona, the mighty Karna, delighting in battle, there did all that Partha had done before. And, O Bharata, Duryodhana with his brothers thereupon embraced Karna in joy and then addressed him saying, 'Welcome O mighty-armed warrior! I have obtained thee by good fortune, O polite one! Live thou as thou pleasest, and command me, and the kingdom of the Kurus.' Kama replied, 'When thou hast said it, I regard it as already accomplished. I only long for thy friendship. And, O lord, my wish is even for a single combat with Arjuna.' Duryodhana said, 'Do thou with me enjoy the good things of life! Be thou the benefactor of thy friend, and, O represser of enemies, place thou thy feet on the heads of all foes."
"Vaisampayana continued, 'Arjuna, after this, deeming himself disgraced, said unto Karna stationed amidst the brothers like unto a cliff, 'That path which the unwelcome intruder and the uninvited talker cometh to, shall be thine, O Karna, for thou shall be slain by me.'
Since the very first time Pandavas come across Karna is when he challenges one of them and takes away his glory, it is but natural for them to dislike him. It is fashionable to show Karna as a tragic hero who was wronged by the society but he was no saint either. In fact Karna also became arrogant with the attention showered on him by Duryodhan as mentioned in SECTION CXXXIV of the same Parva:
The Vrishnis and the Andhakas, and princes from various lands, and the (adopted) son of Radha of the Suta caste, (Karna), all became pupils of Drona. But of them all, the Suta child Karna, from jealousy, frequently defied Arjuna, and supported by Duryodhana, used to disregard the Pandavas.
In view of all this, I think it is but natural for the Pandavas to hate Karna's guts for the rest of his life irrespective of the caste he belonged to. So, in conclusion I would say Karna's caste had nothing to do with the animosity of either Parshurama or the Pandavas.