Concerning your inquiry if someone in Egypt told me that " each " is incorrect , actually no one did but I met the two questions in a book we teach here in Egypt and the given answers are " both " for both questions , so " each " is considered incorrect ( which I disagree with ) [. . .]
Please have a look at the attachment
Hello again, Egyptian2017,
If the circled answers in your attached images are the answers provided by the textbook, it appears that the textbook thinks that "each" should be used instead of "both" in "My parents each have a cell phone" and that "both" should be used instead of "each" in "There are two restaurants by the park and they are both very good."
I wonder, then, if you have misread the answer to "My parents ___ have a mobile phone." The book appears to think that "each" is the only answer. "Both" would also work, but the meaning would be unclear. One might suppose that the mother and father share a mobile phone. And if the plural was used ("My parents both have mobile phones"), they each might have more than one mobile phone.
Thus, "each" is desirable in "My parents ___ have a mobile phone," but "both" is not incorrect. In the other sentence ("There are two restaurants by the park and they are ___ very good), "both" does work very well, though "each" is also possible. The reason "both" is more natural than "each" in that sentence is that there is no reason to emphasize separateness, as there is in the other example.
In the end, then, it appears that your book does not have a misconception that "each" cannot be used in reference to two individuals. It recognizes that that is possible. But it fails to recognize that, where reference to two individuals is concerned, the correctness of "each" does not imply that "both" is incorrect, and vice versa: the correctness of "both" does not imply that "each" is incorrect.
We deal frequently with textbook questions from Egyptian textbooks on the Grammar Exchange, and for the most part it seems as though you have textbooks of high quality. If there is one pattern of error we see, it is that the textbook and test authors often view only one answer as correct. That is only natural. If there were only one answer, English would be like math. But it isn't.
I looked in some books last night to see if I could find an express endorsement of the use of "each" when the reference is to two individuals, and did not find anything, but I know that it is correct. For example, if I were speaking of two boys, I might say, "I gave them each a dollar," to clarify that I didn't give, say, fifty cents (two quarters) to one boy and fifty cents to the other. Passivized, the sentence is:
- They were each given a dollar.