Do parents have to give their children their birth certificate at a certain age?

No, parents do not have to give their children their birth certificate at a certain age.

(Added) However it is wise to do so when the children become adults because they (the "children") will need to show it for various official purposes including possibly, when applying for employment. In fact I had to send my own off only a couple of months ago, to my occupational pension trustees - and I'm now 61!