YES.
There are two levels here. USB3.2 (ALL USB3 systems) uses NINE wires and contacts in the connecting cables, whereas USB2 used only four. If you have "older cables" I would bet they are USB2 with 4 wires. If you use those with USB3 devices, they WILL work, BUT the max data transfer rate will the the USB2 speed of 0.48 Gb/s. To get the USB3.2 max rate of 5 Gb/s or more, you must use the 9-wire type of cable. MORE detail: Any USB3 Type A cable can do the job for USB3.2 Gen1 data rates. If your computer has a Gen2 (up to 10 Gb/s) port you really should use a Type C cable; a Type A cable will work, but may deliver only Gen1 speed. For Gen2x2 (up to 20 Gb/s) you MUST use type C cables.
USB3.2 Type A cables look a LOT like the older USB2 ones. Visually you can tell the difference two ways. The easy way is that MOST USB2 cables have a BLACK plastic support insert in the connector, whereas USB3.2 uses BLUE inserts. Close inspection of a USB3.2 connector will show that, in addition to the four metal contact strips on that support, they have FIVE round dot contacts for the added wires on the support block edge.
The second "level" of this is NOT something you can see - the details of internal design and construction of the cable. There certainly are cheap USB3.2 Gen2 Type C cables that do NOT live up to their design requirements and do not deliver the specified data rates. No way to tell by looking! If you need that quality (or the faster Gen2x2) look for some statement that these cables DO perform that way. And NO cable for that speed claim should have Type A connectors. HOWEVER, there are some cables with Type A on one end and Type C on the other for use with a system that has a Type A socket for what should be a Gen2 port. (The alternative for that situation is to use a Type C cable plus a little adapter block at the Type A port to convert that to Type C.)
There also ARE cable length limits. For normal cables for older USB2 the limit was 5m (16.4 Ft). For good USB3.2 Gen1 and Gen 2 it is 3m (9.8 Ft.); for USB3.2 Gen2x2 it is 0.8m (2.6 Ft.). For distances exceeding these, you must use Active Cables that contain some amplifier system to boost the signal and deal with weaker signals and distortions of waveforms.
My habit is this. For any old devices that were built for USB2 use, I use USB2 cables to connect to older USB2 ports on the system. Such devices cannot do any more than USB2 data rates no matter how they are connected. These include slow devices like keyboards and mice. This keeps any new USB3 port available for USB3 devices. For ANY USB3 devices I would use the correct USB3 cable type AND computer port according to what Gen spec there is for that device.
A side note on speed specs. The basic principle of the USB3.2 speed specs is that the COMMUNICATION subsystem should always be capable of moving data FASTER than the device it serves so that that subsystem never limits the performance of the device. So the speed specs (as above) are the MAX data transfer rates of the USB3.2 communication subsystem in the device or port, and the actual data rate delivered is VERY likely to be slower than that. For example, there are NO classic mechanical hard drives (with spinning disks and moving heads) that can move data faster than about 4 Gb/s because of the speed of movement of their mechanical components, no matter what USB3.2 Gen connection system they use. Most SSD's can get close to 5 Gb/s, and some CAN exceed that, but not as fast as 10 Gb/s (Gen2). Faster devices will come, so Gen2 and Gen2x2 will become necessary sometime soon.