But even this is imprecise and somewhat misleading. The poorest members of society are quite obviously not paying a "larger part" of the debt than the richer members. But the point is that the poorer members are paying disproportionately less as a proportion of their means than the wealthy.
To put it a little simplistically, if there were a flat tax rate of 20% on all earned income, then someone on £10k/year would pay £2k tax, while someone on £1m/year would pay £200k tax. Therefore the wealthier person would pay vastly more actual tax than the poorer person, but the same proportion of his/her means.
But progressive taxation means that the wealthier person actually pays a greater proportion of his/her means in tax than the poorer person. In the income examples above, the poorer person would pay zero income tax from next year, while the wealthier person might pay up to £400k tax.
Thus, the wealthier disproportionately support the poorer. That's the way it should be in a fair, redistributive, compassionate society. But it's rich when people still continue to claim that the poor are bearing a disproportionate burden in fiscal terms (which is explicitly what was being referenced in regard to servicing and paying down the national debt).