A "funded pension" is one where there is a portfolio of stocks, shares, bonds, or other financial assets backing that institution's ability to pay the future pension liability.
Now ... have a think about what that actually means. It means that a fund manager has pieces of paper which gives them rights to the future profits and equity of whatever companies issued those stocks, shares or bonds. It means that the profits made by tomorrow's workers working in those companies will pay the pensions of people who buy into those funds today.
An "unfunded pension" used to mean a promise by a company to pay a pension without there being a stock, share or bond. Some companies acted as if they would be solvent and profitable forever and just issued promises to their employees today that tomorrow's workers would make enough profit to pay out on those promises and afford them a retirement. Remember Robert Maxwell? That was his schtick. Didn't end well for all the Mirror Group pensioners. But his wasn't the only company doing this, and the government at the time, led by John Major (you know ... the Tory?) passed legislation forcing companies to ringfence sufficient funds to meet their pensions liabilities. Contributed to a massive recession as companies had to find zillions to fill all those "unfunded" holes in pensions schemes.
At about the same time, right wing commentators (you know the type ... Tories) started referring to public sector pensions liabilities as "unfunded". Except .... they were never backed by the future profits of any single company, so the distinction between funded and unfunded was meaningless. Public pensions were always backed by the government and funded through General Taxation. This worked pretty well for half a century following the reforms made by Clement Attlee's (Labour) government in 1946 right up until the financial spivs (you know the ones ... Tories) screwed it all up.
Over the last twenty years, the term "unfunded public sector pension" has entered into common parlance, but it is meaningless. The teacher's pension scheme, for example, costs schools and teachers nearly 30% of each teacher's salary, which is paid to the Exchequer. In return, the government promises to pay Teacher's pensions in the future.
Now please explain how any certificate, paid for today on the stock market, issued by a company promising a share of profits tomorrow is different from a certificate, paid for today through pension contributions, issued by the government promising a share of tax revenues tomorrow? They are both promises granted to workers today of future money to be paid when they retire. They both depend on there being enough money earned by tomorrow's workers to keep those promises - and what's more, tax liabilities enjoy a senior position in any company's disbursements than dividends. There is no difference.
The only people claiming that there is a problem are people who want to big up companies and talk down government. And yes, those people are usually Tories.