Dale, it's important to learn to read and understand biographical entries like this one, especially the parts that seem difficult or cryptic.
To help you a bit with these, here is an expanded version:
Perrott, (Sir) James, of county Pembroke, a knight’s son [equitis filius]. Entered Jesus College [Oxford], matriculated 8 July 1586, aged 14. Student of Middle Temple 1590 [that is, he was studying law], as 2nd son of John, of Carew, county Pembroke, knight. Wood says, “a natural [illegitimate] son” (his father was lord deputy of Ireland), of Haroldstone, county Pembroke, knighted in July 1603. Member of Parliament for Haverfordwest 1597-8, 1604-11, 1614, 1621-2, 1628-9, county Pembroke 1624-5. Held office in Ireland. Died without issue [s.p.] 4 February 1636. Brother of John 1580.
Perrott, John, of county Hereford, a knight’s son [equitis filius]. Entered Broadgates Hall [precursor to Pembroke College], matriculation entry under date 28 April 1580, age 15. Student at Gray’s Inn 1583 [that is, he was studying law], as 3rd son of John, of Carew, county Pembroke, knight. Brother of James 1586.
No suggestion in this source that this John is illegitimate.