"Mobile phones have become an indispensable part of our lives."
The sentence as it is phrased is saying mobile phones as a group of things are only one of many parts or aspects of our lives, but they are a part that is indispensable. I might say "My pets are an important part of my life." so the fact that "lives" is plural isn't really relevant.
You could omit the prepositional phrase "of our lives" and the sentence would still be correct grammatically, even though it's not very enlightening. You could also replace it with a different prepositional phrase that uses a singular object, like "Mobile phones have become an indispensable part of our plan for world domination." The prepositional phrase is acting as an adjective describing part.
You would pluralize "parts" if you were talking about how something affects many different aspects of your life or the way you live, for example, "Almost all parts of my life have been changed by technology."