No, you don't need to order any certificates at all. That is, if you are looking for births and deaths.
You can search the GRO indexes for free here (you may need to create a free account first though)
https://www.gro.gov.uk/gro/content/certificates/indexes_search.asp
So, for example, if you know that a Bob Blair married a Sally Booth (just random examples) then if you put in both surnames you will be able to find out the names of any children that they had. You can search up to five years at a time.
The only problem is that if it is a very common surname then it will give a lot of other results as well as the one that you are looking for.
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So, it's simple to find children on the GRO index when you already know who the parents are.
But, if you want to find out the surnames of the parents of somebody who was born you have to use Ancestry or some other website to get that information.
The reason is that the GRO do not give out mother's maiden name for anyone born in the last 100 years (so, after 1925). But if the person you're looking at was born in 1925 or earlier then you can still use the GRO to search for their parents surnames.
However, this information is available on Ancestry and other websites.
So, if you are looking for the parents of a particular person then on Ancestry (or the GRO if they were born 1925 or earlier) you put in the name, rough date of birth and rough location of birth.
It will then bring back a list of everyone with that name who was born in that area in that time period.