It is understood thousands of parcels are still waiting to be delivered at the main Brunswick Road depot, almost a month after the industrial action ended.
As staff battle to clear the delayed mail, hundreds of residents are being sent postcards suggesting they collect undelivered items.
Industry regulator Postwatch today hit out at the decision, saying customers who paid to have their packages delivered would at the very least have expected the company to try to make a delivery.
Residents in EH7 and EH8, covering central Edinburgh and the Leith, Restalrig, Southside, Newington and Canongate areas, began receiving the postcards at the weekend.
The postcard also states residents have the option of phoning up to arrange a delivery or paying to have it delivered to a post office.
One postal worker, who asked not to be named, said: "People have paid for these items to be delivered, but Royal Mail has decided it is not going to bother.
"Many people will have had packets sent by post as they are elderly or infirm and cannot get out. It isn't fair that they be asked to come and collect them."
It is thought attempts to clear the backlog of mail in Edinburgh were hindered by large deliveries of parcels from depots in England, which are also recovering from the effects of the strikes.
The dispute over pay and working conditions, which led to an estimated 30 million letters going undelivered, came to an end on October 12, two days after the last strike in Edinburgh.
A spokesman for Postwatch said: "If someone has paid for a parcel to be delivered, they would at least expect the Royal Mail to attempt to deliver it. It seems unfair that this long after the strikes ended that the customer should still be losing out.
"There have obviously been difficulties clearing the huge backlog and I know the Royal Mail has done a lot to try and get that delivered, but we have not heard of this being done in any other part of the country. I do not see why an Edinburgh depot, even one as busy as Brunswick Road, should still be having these problems."
A Royal Mail spokeswoman insisted people were only asked to come in to collect packages as one option. She said customers could also phone the depot and ask for the item to be delivered, or have it sent to the nearest Post Office.
"The vast majority of packets have been collected or delivered," she said. "Cards only went out on Friday. This was a measure to get parcels to our customers over one weekend and was caused by other parcels arriving from elsewhere in the country.
"Edinburgh has worked well at clearing the backlog, but given the size of our network we are still seeing some backlog mail arrive with us from down south."