Quick answer, unfortunately you have to buy Birth Certificates & sometimes marriage cert.
Starting with you, your [long form] B/C should [if he is named, sorry not being rude] show your fathers name & mothers name with formerly Smith, Jones, Bloggs etc [if they are married]. From this you can look for a match in marriages of your parents, the M/C cert will show the bride and groom their ages at time of marriage, plus each one's father [again assuming they are named or were known]. You now have another piece of the jigsaw towards your parents B/C, your fathers name off your B/C & his marriage cert plus his fathers name off his own M/C & the same with your mother. Now you can look for all the Tom Bloggs for a particular year in a particular area and hope there are not loads. Warning though sometimes we have to guess as to where a person was born but as with my own experience found much to my surprise that I am one of 4 we have lived my whole life in Stoke On Trent, I always thought we were all born here but was shocked to find my brother had been born in Plymouth. So its a B/C followed by a M/C followed by a B/C or two if you trace each branch of your family. Use your local Registrar if your lucky that people live local & or the General Registry Office at Southport for certificates other sites will cost a fortune. When you get back you will find the Census records of 1841, 1851, 1861, 1871, 1881, 1891, 1901 & 1911 invaluable but they don't show mothers maiden names. Hope this has been of some use & good luck with your search