Is reading supplications for one's parents from a computer device, when it is constantly running, considered an innovation (bid'ah), or is that permissible?

There is no harm for a Muslim in asking Allah Almighty and in supplicating Him frequently at any time, as long as there is no Shar'i (Islamically forbidden) impediment in the supplication. So, if a specific supplication appears on the device screen every time the device is opened to remind one of the supplication, it is a good thing. However, it is better not to adhere to it always and to add other supplications or to engage in recitation at times, so that this does not fall into the category of innovative additions (bid'ah idafiyyah).