Perpetual American Bank announced yesterday it has made $10.6 million in profits by selling several pieces of real estate, including a downtown Washington parking lot.

The transactions are expected to add about $1.56 a share to the fiscal 1985 earnings of Perpetual American, which earned $1.28 a share during the first half of the year, ending April 30.

The largest part of the $10.6 million gain -- $7.4 million -- came from the sale of a 16,000-square-foot parking lot adjacent to Perpetual's former headquarters at 500 11th St. NW. The land was purchased by Manufacturers Life Insurance Company of Toronto.

The remainder of the $10.6 million gain -- $3.2 million -- came from the sale of Perpetual American's branch office building at 8700 Georgia Ave., Silver Spring, and from the sale of an interest in an unidentified downtown office building. Sale prices were not revealed.

Still up for sale by Perpetual American is the Washington building, located at New York and Pennsylvania avenues NW. It has been on the market for about $35 million.

Meanwhile, the institution announced that its forthcoming issue of $35 million in cumulative convertible preferred stock will be handled by Alex. Brown & Sons and Merrill Lynch Capital Markets. The bank expects to sell $3.5 million shares at $10 a share.