The simple answer is no.
Civil Registration Indexes do not include the names of parents. From the middle of 1911 onwards, they do include the maiden name of the mother.
If the children were baptised you should be able to see the parents married names (i.e John and Jane Smith). Many children though were not baptised. Findmypast does have an extensive array of parish registers, as does FamilySearch and Ancestry.
The only way for certain that you will get the parents names is to purchase birth certificates at a cost £9.50 each from the GRO. When you order certificates it is possible to include on the order form the names of the parents that you are looking for, and if the entry does not include the identified parents you get a full refund.