Former party chairman Mr Parkinson made his former secretary Sara Keays pregnant following a 12-year affair. He quit in October 1983 when details emerged.
Mrs Thatcher's private papers include documents that appear to confirm she was grooming him as her successor.
They also include previously unseen resignation letters, in which she describes the situation as "tragic".
Two handwritten notes show the then prime minister intended to promote Mr Parkinson, who became Lord Parkinson, to foreign secretary before the scandal broke.
In one she wrote simply: "F.S. C.P."
The move has long been speculated about, but the notes are seen as the first solid evidence.
Chris Collins, from the Margaret Thatcher Foundation, is the only historian to have studied the archive. He said such a move would have catapulted Mr Parkinson up the party hierarchy, effectively anointing him as her heir.
"It's only speculative that she considered him to be her successor but it would appear that she was very serious about him indeed," he added.
"He had certainly risen very fast and she knew that by promoting him in this way, she would create an atmosphere around him."