The sentence means that Della was embarrassed (cheeks burning) by the unspoken (silent) suggestion (imputation) of stinginess (parsimony). Della and Jim had very little money and so they had to make every penny count and to some that might appear that they were being stingy rather than simply being frugal. The suggestion that they had to be so frugal was embarrassing, thus causing a blush on her cheeks.
In the story, it is this lack of money that causes Della and Jim each to part with her/his most loved possession in order to buy the other a special gift. Of course, the irony in the story is that the gift each purchased for the other was a gift that would have adorned the possession that each sold.