I have 33 girls on my team right now. I've had three leave in a very awkward manner this season. The parents of all three are not big fans of my coaching methods but I really don't care. I have to think of the team first. That is your job as the coach of the team.
The kids that come to practice and cut corners are extremely bad for the culture of the team. If I see them slacking off I usually send them home and ask them to really think about whether they want to be on the team or not before returning. They either quit or change their behavior. When they quit the other girls tend to take their training and team more seriously. I always see a change in attitude and it's generally a positive one.
You can't make everyone like you and if you do you're probably not a very good coach. Just the fact that this parent is not fond of you tells me you're probably doing the right thing. You should take pride in that.
I love the kids who put in a big effort but have no talent. They are the best because they are doing it to better themselves and inspire other team members. Make sure you give those kids the extra attention they deserve. If you have the right attitude on your team they will all get faster and it will become the norm to work hard.
The culture of your team is the most important thing you can build as a coach. Kids with talent who are slackers are absolutely the worst members of your team. They show everyone on the team they can have some measure of success without working hard. Then everyone thinks they can train like them.
Another method that tends to work is not allowing a kid to race when they are slacking off. I tell them and their parents if I need to that they are not ready to race and will be allowed to race when they show they can handle the training. I tell them I don't feel it is safe for them to compete without a serious risk of injury.
Anyway, I hear you. I have my share of crazy parents. They are not rational when it comes to their kids and will defend them whether they are right or wrong. You just have to be more thick skinned and consider that they are "crazy" as you put it. Why should you ever value the opinion of a crazy person?