My identical twin sister and I were born here in Lima, Peru in February 23rd, 1987 to a British Father and a Peruvian Mother. However, they never got married and our father died in October 04th, 1995. My father was born in England, as a consequence, I believe that he was a British citizen by birth, and so, we can be British citizens by descent.
I have read that marriage between our parents is no longer required, so I would really appreciate it if someone could please confirm this information and let me know if we are eligible for the British Passport.
In that case, what should we do? Which will be the process to obtain the British Passport?

Your parents have never been legally married... this answers all your questions.

He was able to report births of his out-of-wedlock children when he was alive, and he failed to do it. Sorry...

Thanks for your answer, but I believe that their marriage isn't a requirement anymore.
Besides, his name and signature appears in our Birth Certificate and when he went to the British Embassy when we were born, he was told that he couldn't register us since he was not married by that time.