Can a device designed to plug into a USB port also plug into a serial or parallel port?

Generally, not easily. You can buy adaptors, but they don’t alwas work right.

You can get adapters, but many USB devices require power that comes through the USB port. Also, serial and parallel ports are extremely slow in comparison with USB. If you need more ports, you would be better off getting a USB hub that will allow you to plug multiple devices into one USB port. Think of it like an electrical power strip. A single USB port can support 127 devices.
Here’s a article on the port differences.

I am fairly certain that there are adapters for USB to serial, and USB to parallel. They often require external power. I am not sure of the other way round, serial to USB or parallel to USB. For the most part, printers and scanners etc that used serial and parallel ports are attrition out. Soon adapters will no longer be necessary.

Could be the biggest issue would be finding a device driver that would communicate with that device on a port it was not designed to be connected to…

Keyboards and mice… they pretty much could work. The mapping of signals and meaning are straightforward. And some of those still come with a USB to PS/2 connector, for example – and a disk with a serial/ps2 driver on it.

And as mentioned above, datarate is also an issue for most devices. USB2 > > > serial rs-232 or parallel ieee 1394 ports.

Contemporary printers or a webcam, forget it.

And there’s the matter of two-way communication, which USB can do easily.
Only the crudest of handshakes is available over parallel and default serial port drivers don’t expert any or much feedback from the peripheral gadget.

Dabbler is correct that it is a driver issue and most likely won’t work even if can plug them into each other.

USB ports have four “pins”; two for power, and two for a high-speed serial data connection. As many USB devices require power and lack external power connectors since they expect the USB port to provide power, you already have one issue.

The other is that data rates truly do take a hit. To give you a hint, here are the bandwidths:

USB 2.0 (currently the most common) = 480 Mbits/sec
USB 1.0 (a decade old and rarely seen now) = 12 Mbits/sec
Parallel port = 1 Mbps (typically)
Serial port = 230.4 Kbits/sec = ~0.23 Mbits/sec

As you can see from the numbers, transfer rates are hundreds of times faster with USB. Some devices can handle that slow a data rate while others cannot, and others will do so with issues. A disk drive will read/write sssssssssssssssssssslllllllllllllllllllllllllllllooooooooooooooooooowwwwwwwwwwwwwwwwwlllllllllllllllllllllllllllllllllyyyyyyyyyyyyyyyyyyyyy. A large file that I could move in a about a minute on USB 2.0 will take me well over half an hour on USB 1.0, about eight hours on a Parallel port, and about a day and a half on a serial port. You do NOT want to use such a connection for data transfer of any notable size. It would work, but not well

Something like a video camera uses enough bandwidth that it would just fail outright. A mouse may act funny; sluggish, and jumpy. A keyboard wouldn’t care unless you type a bazillion words a minute.

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TL:DR version – Just because you can do something, that doesn’t mean you should.