How do I read my child's text messages?
If you have Verizon Messages installed on your child's phone and he uses it as the default messaging platform, you could log in to his MyVerizon account with their credentials in order to view the messages. Of course you would need his userid and password for their MyVerizon account in order to do so. Depending on the age of the child this may or may not be easy to acquire.
You cannot read the actual text message. You can control the numbers that texts are sent/received, calls are made to and from. Text message content is not available to view at will. To obtain this information, would require you to obtain a court order.
Edit: The only way to read the actual text content, is by having access to the device itself, or with a court order. Being the account owner does not change this.