A child who is allowed to be disrespectful to his parents will not have true respect for anyone.
Why would I take issue with this? Obviously I think children ought to respect their parents, and obviously I agree that it's a good thing for children to have respect for other people, as well. These are certainly desirable things. But often when we read or hear something that seems to be about uncontroversial values we all share, we lapse into a kind of warm glowy agreement with it, and don't look closely enough to notice that what it's actually saying is, well, wrong.
There are, after all, children who are born to disreputable and irresponsible parents, who still somehow manage to grow up into decent and respectful human beings despite (and sometimes as a reaction to) their awful role models. It is entirely conceivable that such a child might not respect her parents at all, having been disappointed in them too many times to extend them the benefit of the doubt, and yet still have true respect for others. So the claim is empirically false, but it goes deeper than just the occasional exception-that-proves-the-rule.
The real problem I have with this claim is that that's just not how respect works. Respect is not a behaviour but an attitude, a recognition that other people exist, that they have their own perspectives which we are not well-situated to judge, that they have their own concerns and interests, and most of all that they matter. Once one has that basic element of respect, all sorts of respectful behaviours just naturally emerge from it. When we respect each other, we will defer to each other's preferences and expertise, or at least take them into account. We will choose our words politely, so as to avoid giving offense, because respect means that a person's preference not to be offended matters to us.
Now, what does it mean, then, to "allow" a child to be disrespectful to his parents? From my perspective, this is an almost meaningless question; how can you allow someone to have an attitude or belief about your inherent worth as a human being? Indeed, the very question is itself disrespectful, insofar as it disrespects the basic sovereignty of the child's own mind.
What one can allow (or disallow) is behaviour, and yes, it's possible to demand politeness and deference, and to punish their absence. And by all means, as parents we certainly ought to attempt to teach our children how to behave politely, towards their parents as well as towards anyone else. A respectful attitude can emerge from this, as we model the practice of considering others. But if we are focused on demanding that our children show us polite deference as their parents, we might well end up modelling an authoritarian self-centeredness that undermines the foundations of true respect.
And so I would say that, as superficially profound as the Rev. Graham's quote sounds, it's actually backwards. Teach your children what respect is, not by demanding it from them, but by modelling it for them. Demonstrate respect by showing it to everyone.
You cannot make your children, or anyone else, respect you. What you can and should do is strive always to be worthy of that respect; whether you get it or not is not up to you. And if your children live by the same principle, you'll have done your job well.