Hi Trevor,
No luck on the parents name front I'm afraid

Only one mentioned a paarent, and the name was not transcribed! There were two different locations for baptisms though, which might help a bit. Entries in full were:
Hackinsall; Marye d. of ...... Bamber of Hackinsall, 11 Dec 1625 [presumbaly name was unreadable! How frustrating]
Preesall; Paup. John Bamber, 9 Sep 1627
Hackinsall: James Bamber, 14 Apr 1627
Preesall; Elizabeth Bamber, 23 Jan 1627 [1628 by modern calandar]
Hackinsall; John Bamber, 27 Jan 1632 [1633 by modern calendar]
Also the following burials around that time
Hackinsall; ........... Bamber, 4 Oct 1630
Preesall; Margaret Bamber, 20 Jun 1636
And marriages
Hackinsall; John Bamber to Elizabeth Howell, 12 Oct 1616
Hackinsall; William Bamber to Ellinge Barker, 16 Feb 1624 [1625 by modern calendar]
Stalmine; John Styham to Margaret Bamber, 18 Aug 1622
There were some later entries too for around 1720, and the book linked the names Bamber, Bambridge and Bembridge. But all yours were Bamber.

Barbara