American Security Corp., the parent company of the city's second-largest bank, yesterday reported a 13.2 percent profits rise for the 1980 third quarter.

American Security said profits for the quarter before securities transactions were $6.4 million ($1.79 a share), up from $5.6 million ($1.55) for the third quarter in 1979.

For the year the bank said profits are up more than 17 percent, with profits totaling $18.2 million ($5.05), compared with profits of $15.5 million ($4.25) for the same nine months last year.

Total assets as of the end of the quarter rose to $2.5 billion, up from $2.2 billion at the same date in 1979. Loans at American Security rose from $1.3 billion at the end of last year's third quarter to $1.4 billion as of Sept. 30 of this year, and total deposits rose from $1.98 billion, and increase of $197 million over the figure at the same time last year.

American Security Chairman W. Jarvis Moody attributed the quarter's results to the bank's ability to maintain "good spreads" between the bank's loans and the money it borrows.

"All in all, even though the fourth quarter may not be as strong as the second and third quarters, we expect to finish the year either on or ahead of target," Moody said.

But a major Virginia bank yesterday issued a far different profits report. Virginia National Bankshares Inc. yesterday reported a sharp 13.3 percent profits drop, as earnings fell from $5.4 million (76 cents a share) for the third quarter last year to $4.7 million (67 cents) for the same quarter this year.

For the first nine months of 1980, Virginia National reported profits before securities transactions of $16.7 million ($2.40), up 6.8 percent over 1979 nine-month-profits of $15.6 million ($2.30).

"While it is anticipated that market forces may prevent our achieving 1979 forth-quarter results, we are confident that 1980 earnings will approximate or exceed 1979's record," and C.A. Cutchins, Virginia National chairman.

Total assets rose to $2.62 billion at end of the third quarter this year, from $2.52 billion on the same date last year. Virginia National's total deposits rose from $2.04 billion at the end of the 1979 third quarter to $2.12 billion at the close of the same quarter this year.