They do not need to cover the PSEC shares before the rights offering x-date, but if they don't, they will be short the rights when they are issued as dividends, and they will need to buy the rights in the open market to cover their short positions.
The shorts in PSEC will need to buy the rights (to buy PYLD) because those rights are treated as dividends. With cash dividends, shorts are forced to pay by simply debiting their account, but the shorts here don't have any rights sitting around while they are required to deliver them. So they will be forced to buy them in the open market. The entire 25 million of them.
Your article and all of the comments seem to assume that the cost to exercise the rights will be equal to the asserted value of the CLOs. It is quite possible that the cost to exercise will be lower, thereby providing a nice incentive to exercise them.