There is no एक, so shouldn’t it be “Neha is writing the book” ?
Sometimes the एक can be omitted if the meaning is clear. That is what's happening here.
then why was it needed in "Peter's father is writing a book". I left एक out and it was wrong.
The book is not specified so the “ek” is unnecessary and as @zeebo7 said the word can omitted since it is rarely used because most of the times the meaning is clear.
I wrote "Neha's writing a book" and it said I had a typo. I think Duolingo needs to accept more casual language in its answers.
"Neha's writing a book" should also be accepted.