Respect is given; trust is earned. Respect and trust come in different levels.
To respect is to hold a favorable opinion of someone; at the most basic level, every living thing deserves respect for the fact that it lives. People all deserve respect for their thoughts, however different from yours they might be. There are higher levels of respect, a fact which is exploited in many ways in civilized society. For example, religious hierarchies, where the local clergy (ie pastor, priest) is given higher respect (and expected to live up to it) than the church members; the next-higher-level clergy (ie, bishop)is given more respect than the local person; the level above that is given even more respect, and so on up to the top, where the level of respect might reach veneration.
To trust is to rely upon someone or something. Trust is earned, and the level of trust builds over time as the trusted thing or person shows repeatedly he/she/it is trustworthy (worthy of our trust). We trust our car to get us from point A to point B. We trust that the food we buy at the store is safe to eat. We trust that other drivers will stop at the red light as we pass through on the green. We trust day care providers and schools to provide appropriate environments for our children.
One can trust someone without respecting him/her, and one can respect someone without trusting him/her. For example, one can trust a politician to lie (which does not involve respect); and one can respect the President of the USA without trusting the programs and policies he/she espouses.
To have true intimacy with someone, both respect and trust are required. To function as separated parents of a child, respect is necessary and trust is a plus. To grow up emotionally healthy, the child must know that each parent respects the other. Even so, "trust but verify" that the child is receiving proper care in a healthy environment. Same goes for day care providers and schools, "trust but verify".