The answer is not just yes or no unfortunately, it's not that simple.

Yes the card could work in your board, but its requirements must be met.

The ASRock ASRock SATA3 Card, like any SATA III PCIe add on card, must be used in a PCIe 2.0 slot or you won't get the maximum speed from a drive used with it.

Your P43DE board has one PCIe 2.0 x16 slot. The PCIe x1 slots operate at PCIe 1.1 speeds. That is 5Gb/s for PCIe 2.0, and 2.5Gb/s for PCIe 1.1, per lane.

If you used this SATA III card in the PCIe 2.0 x16 slot, the card will operate at its full speed. But when used in a PCIe 1.1 x1 lane, it will be slower than the SATA II ports on your board.

If you want to use this card with your OS/boot drive, older boards like yours may not allow you to boot from a drive used with these cards.

If you wanted to use this card with a SATA III SSD, I suggest using an Intel SATA II port. You will still get the majority of the speed benefits of a SSD, except for the ~500MB/s large file sequential read and write speeds.

If you planned on using this card with a SATA III HDD, don't bother. The number of so called "SATA III" HDDs that can even reach the maximum speeds of SATA II is exactly zero.