Royals keeper Grace Moloney foiled Mead before Lois Roche put Reading ahead with a deflected free-kick.
Mead rounded Moloney to level after the break and then had an effort cleared off the line by Kirsty McGee.
Sunderland keeper Rachael Laws made a series of saves as Reading earned their first ever point in the top flight.
Sunderland Ladies: Laws; Holmes, Bannon, Williams, Sjoman; Furness, Chaplen, Staniforth; Joice (Ramshaw 46),  Mead, Johnston (Kelly, 46).
Substitutes not used: Gunn Olsen, McDougall, Roche, Beer, Potts.
Reading Women: Moloney; Jane, Bartrip, McGee, Rowe; Hines, Roche, Estcourt (Bruton 58); Ward (Jones 90), Boho Sayo, Follis.
Substitutes not used: Earps, Cox, Scott, Horwood, Walkley.
Attendance: 699.
Referee: Glen Hart.
Last season's top scorer Beth Mead opened her 2016 account as Sunderland fought back to deny Reading their first ever Women's Super League One win.