There isn't a rule against traveling with a stroller, so I don't think that TSA will prevent it. They might question it being there with no kids in sight, but it's a pretty simple matter to tell them that your children are still getting through baggage check, and you're helping out. (Note that the airline itself will probably insist there be a child in evidence when the stroller is gate-checked. Strollers are only a free item eligible for gate-check when there is a young child traveling. If there is no child traveling in the party, strollers have to go onto the plane as checked baggage, and some airlines will charge a fee for them in those circumstances.) If you do opt to carry the stroller for them, remember that it must be fully folded to go through security; as it must pass through the scanner if it will fit, and if it is too large to do so, it will be a slow process to get it hand-wanded. Make sure you know how to fold and secure it, and that someone who is strong enough is responsible for lifting it on and off the belt.
Depending on which airport you are going through, your children might object more than TSA will. Strollers are tremendously useful for keeping young children under control and where they are supposed to be when moving through an airport. Based on my experience, a place like DEN for instance, or ATL, would be a bear without one if the kids are preschool-aged. Looking back on my time travelling with young children, if I had had able-bodied grandparents traveling with us, I think I would have been more grateful to have them take responsibility for moving the children themselves -- or at least one of them. Little kids somehow just seem to instinctively be on better behavior for grandparents when in new and exciting places, and that leaves the parents free to just deal with all the stuff without having to worry so much about what the children might get up to.