This further information is provided courtesy of Neville Hurworth:
"Aske Hall is of great antiquity. In the Doomsday Book it is described as ASSE, the manor of Tor, a Saxon, before the Conquest. Then it belonged to Whyomar, or Whyomere, Kinsman and Sewer (footnote says "An Officer who served the dishes and arranged the table") to Allan, the first Earl of Richmond. Whyomar's descendants flourished here for more than 500 years under the name of Aske. The Askes were the first founders of Easby Abbey and great benefactors to other religious houses......(Sentence cut here)... The line of Aske ending in females, Elizabeth one of the daughters and coheiress of Rodger de Aske, carried this Estate in marriage, about the year 1530, to Richard Bowes, a third son of the Streatlam family in the County of Durham. Their eldest son, Sir George Bowes of Aske, Knight Marshall, became heir male of the whole family of Bowes, upon the death of his cousin, Sir George Bowes, Streatlam, in 1556, and their second son, Robert, was settled at Aske, and married as his second wife Eleanor Bowes, who left several charities to the town of Richmond. Some time before the year 1640, Philip, Lord Wharton, purchased this property from Sir Talb?ot Aske (I am guessing the letter "b" in "Talbot" which has a blemish in the paper here which obscures one letter), and in this family it continued till 1727 when the large sums expended by Philip, Duke of Wharton, and by his father the Marquis, in electioneering matters, so burdened his estate that a decree in Chancery vested it in trustees for the payment of his debts. In the same year, the trustees sold Aske to Sir Conyers D'Arcy, who improved the lands, repaired the hall and made it his chief place of residence. At his death in 1758, he bequeathed Aske to his nephew the Earl of Holderness, who sold it in 1762 to Sir Lawrence Dundas, Bart. Sir Lawrence made great additions to the hall, gardens, stables and out-offices. An old border tower, the only remnant of the Askes, was left uninjured, boldly towering above the hall and new buildings"
Source: "The Zetland (Dundas) Archive"