The exact rules are set out in the various Acts of Parliaments many of which are transcribed on my Acts site at
http://freepages.genealogy.rootsweb.com/%7Eframland/acts/actind.htmThe 1836 Act covers the relevant period of your question, but it should be noted that the requirement was for the registrar to ask for the information at that time rather than the parents being obliged to provide the information.
Extract from 1836 Act -
XVIII. And be it enacted, That the Registrar General shall furnish to every Superintendent Registrar, for the Use of the Registrars under his Superintendence, a sufficient number of Register Books of Births and of Register Books of Deaths, and of Forms for certified Copies thereof, as herein-after provided, at a reasonable Price, to be fixed from Time to Time by One of His Majesty's Principal Secretaries of State, the Cost whereof shall be borne by the Union, Parish, or Place in or for which the Superintendent Registrar is appointed, and shall be paid by the Guardians or by the Churchwardens and Overseers (as the Case may be), out of monies coming to their Hands or Control as such Guardians or Churchwardens and Overseers, to the Registrar, and shall be accounted for by him to the Registrar General ; and every Registrar shall be authorized and is hereby required to inform himself carefully of every Birth and every Death which shall happen within his District after the said First Day of March, and to learn and register soon after the Event as conveniently may be done, without Fee or Reward save as herein-after mentioned, in One of the said Books, the Particulars required to be registered according to the Forms in the said Schedules (A.) and (B.) respectively touching every such Birth or every such Death, as the Case may be, which shall not have been already registered, every such Entry being made in Order from the Beginning to the End of the Book.
XIX. And be it enacted, That the Father or Mother of any Child born, or the Occupier of every House or Tenement in England in which any Birth or Death shall happen, after the said First day of March, may, within Forty-two Days next after the Day of such Birth or within Five Days after the Day of such Death respectively, give Notice of such Birth or Death to the Registrar of the District ; and in case any new-born Child or any dead Body shall be found exposed, the Overseers of the Poor in the Case of the new-born Child, and the Coroner in the case of the dead Body, shall forthwith give Notice and Information thereof, and of the Place where such Child or dead Body was found, to the Registrar ; and for the Purpose of this Act the Master or Keeper of every Gaol, Prison, or House of Correction, or Workhouse, Hospital, or Lunatic Asylum, or public or charitable Institution, shall be deemed the Occupier thereof.
Cheers
Guy