Let us first assume that, in effect, the parents are compelling businesses to give the donations. In theory, the gemara says that one who pressures others into giving tzedaka is greater than the one who actually gives it.[1] However, this is assuming that the donor was obligated to give that tzedaka and was reluctant to do so. It should be noted, though, that one who pressures another who truly cannot afford to give tzedaka into contributing is liable to be Divinely punished.[2]
In any case, parents have no right to compel a storeowner to donate to the institution of their choice. Who says that he doesn’t have other causes to whom he would like to give his donations? Who says he hasn’t given enough tzedaka already? If he is not Jewish, he doesn’t have an obligation to give tzedaka at all. What qualifies the parents to make these decisions, especially when they have personal interests in the matter?
Practically speaking, though, it’s hard to believe that parents actually have the ability to coerce a merchant to donate. They could impose some type of psychological pressure, but the matter ultimately would remain the merchant’s decision. Having said this, any type of significant, negative pressure would be inappropriate, and in some cases, could be forbidden.
However, in the type of scenario you refer to, it appears that the incentive to contribute is usually positive. In other words, the storeowner is interested in creating or maintaining goodwill and respect with the community of potential customers. Just as customers can ask for courteous treatment, they can also ask for generosity. In either case, the decision remains with the business owner. The contributor often benefits from his donation by being publicly acknowledged. The actual cost to the donor is less than the retail value, and it may be possible for him to get a tax break. If these are the conditions, there should not be any halachic or ethical problems in soliciting the donations. Of course, the solicitor should conduct himself in a manner that preserves the honor of a Torah lifestyle.
Regarding the parents, they have more of an obligation to assist the school that educates their children than others do. However, given the high cost of Jewish education, it is unrealistic for many of them to contribute more than the tuition they already pay.