If this student has failed grade 4 and 5, why on earth would you even consider putting them in for Grade 6, pushy parents or not! Does this student have a weird belief that just by entering for an exam they must therefore be the standard of the grade regardless of whether they pass or fail? It's like saying, I'm grade 6 standard now because I was able to open the book and practice the exam pieces!
My advice would be to put your foot down to this student and take control! If they had failed only Grade 4 or only grade 5 then maybe they could pass Grade 6. But if they have failed BOTH of them, they obviously weren't up to the required standard, suggesting that the very same thing could easily happen again.
If this was one of my own students, with two failed exams behind them, I would insist they retake the last failed exam before moving forwards.