I've tried looking through all three vol.s of Ormerod's The history of the county palatine and city of Chester: .... but could not find (unless I overlooked it) anything exact on Eleanor's parents.
The history of the county palatine and city of Chester: .... Vol. II.
https://archive.org/stream/historyofcountyp02orme#page/350/mode/1up
Sir Philip Egerton obitt 17 July 1563 m. Eleanor, dau. of sir Randle Brereton, of Malpas, marriage settlement dated ante 26 Hen. VIII. deed 3d Nov. 1567; buried at Little Budworth, Nov. 5.; ch: Eleanor (m. sir Randle Brereton), John (m. Jane Mostyn) Egerton.
The history of the county palatine and city of Chester: .... Vol. III.
https://archive.org/stream/historyofcountyp03orme#page/51/mode/1up
Sir William Brereton, of Brereton, kt. chief justice and lord high marshal of Ireland, bur. Feb. 4, 1541, at Kilkenny in Ireland m1. Alice Savage., 2dly Elenor, dau. of sir Ralph Brereton of Ipstones, widow of . . . . Egerton.
These two must be different Elenor's as Philip Egerton is listed as d. 1563 and Sir William Brereton as 1541, so the Elenor Brereton who married Sir William was not the same as the one married to Sir Philip Egerton.
To make a guess I would say Sir William's wife would be the elder Elenor, dau. of Emma (Carington) & Randle Brereton, & probably a widow of some other Egerton.
And Sir Philip Egerton's wife Elenor, dau. of Eleanor (Dutton) & the younger Randle Brereton.
But this would be just a guess and may not be correct. Maybe the Randle's are completely different then these two?