Yes, it is possible that such occurrence can damage a HDD. However with just a straight single power off, such as unplugging it, it's unlikely to cause damage. What is likely to cause damage is the flickering on/off/on/off that often happens during a power outage caused by a brownout or storm.
HDDs use the centrifugal motion of the platters to generate electricity after losing power. This electricity is needed to park the heads safely onto the parking ramp or parking zone which is coated so as to prevent stiction or other damage to the heads. If the platters are spinning up at full speed, it'll have more than enough juice to park the heads safely. However, if the power goes off and back on in repeated succession (as it often does) the platters may have just started to spin back up in the last power on and haven't yet reached full speed. So the spinning of the platters may not have enough potential energy to power the parking sequence. In such a case the heads often become fused to the platters.
We see this all the time after storms here at the data recovery lab. Fortunately they are usually always easy complete recoveries.
If you use a UPS though, even if it dies before shutting the computer down or power being restored it's unlikely to cause damage. The UPS will do just a single clean shut off.