Yes, it is perfectly permissible for anyone who comes to Makkah for the Umrah to perform more than one Umrah on the same journey. However, he must start his second Umrah from a point outside the Haram area. The nearest is the mosque at Taneem. When the Prophet (peace be upon him) performed his pilgrimage, all his wives were with him. He advised his companions to start with the Umrah and wait for the pilgrimage days to start on their major duty. However, Lady Aishah was in her period and was unable to perform the Umrah on her arrival. When she complained to the Prophet that she would go back having performed the pilgrimage only while others did both duties, he told her brother to take her to the nearest point outside the Haram area. He took her to Taneem where she started her ihraam, or consecration, again and came back to perform the Umrah. Thus, the requirement to start outside the Haram area is clear, even when the second Umrah is on behalf of someone else.
Should we perform a second Umrah if we are in Makkah? Unless we are doing the second Umrah on behalf of someone else, then we should not. It is preferable to do the tawaf as frequently as we wish while we are staying in Makkah. If you are doing the second Umrah on behalf of a deceased parent or relative, then this is perfectly appropriate. You have to go outside the Haram area to start your journey again, entering in the state of consecration there and coming back to perform your tawaf, sa’ie and cut some of your hair or shave your head. It should be explained that you must have done one Umrah before you can do it on behalf of someone else. Moreover, it can be done on behalf of someone who is either unable to do it because of a long lasting illness or a deceased person. If your parents are healthy but cannot afford the journey, you cannot do the Umrah on their behalf. If you want to be very dutiful to them, then you should invite them to come over, and pay their expenses, if you can afford that. You will earn rich reward from God for doing so.