Standard Oil Co. (Indiana), the sixth-largest U.S. oil company, reported yesterday that its third-quarter operating profits rose 17.6 percent, primarily on higher exploration, production and chemical earnings.

And Standard Oil Co. (Ohio), the 13th largest, logged an 8 percent gain in the third quarter. Sohio cited an increase in Alaskan crude oil prices and stepped-up sales of refined petroleum products.

Indiana Standard had third-quarter operating earnings of $528.3 million ($1.81 a share) compared with a net profit of $448.9 million ($1.53) a year earlier. A one-time gain of $80 million from the company's sale of its 62.2 percent stake in Cyprus Anvil Mining Corp. Ltd. boosted final net for the latest period to $608.3 million ($2.08).

Third-quarter revenues climbed 23 percent to $8 billion from $6.5 billion.

Nine-month profits were up 1 1/2 percent to $1.53 billion ($5.27) from $1.51 billion ($5.16) in the same period last year. Revenues advanced 18.9 percent to $23.9 billion from $20.1 billion.

Sohio's third-quarter earnings climbed to $479 million ($1.92 a share) from $433.4 million ($1.77) a year earlier. Revenues rose 20 percent to $3 billion from $2.5 billion.

Nine-month earnings were $1.47 billion ($5.98), up 10 percent from $1.34 billion ($5.44). Revenues rose 12 1/2 percent to $9 billion from $8 billion.

American Airlines Inc. said yesterday that earnings in the third quarter fell 42 percent from the same 1980 period, while Trans World Corp. said its earnings were off 38 percent. Both complained that profits suffered as a result of the strike by air traffic controllers, which began Aug. 3.

Trans World Corp. earnings in the third quarter were $74 million ($3.21 a share) on revenues of $1.47 billion compared with earnings of $119.7 million ($6.99) on revenues of $1.40 billion a year earlier. Without an extraordinary $49.4 million gain, earnings for the third quarter of 1980 were $70.3 million ($3.97).

Nine-month earnings were $73.5 million ($3.02) on revenues of $4.03 billion compared with earnings a year earlier of $82.3 million ($4.08) on revenues of $3.75 billion. Without the extraordinary gain, earnings for the first nine months of last year were $32.9 million ($1.04).

American reported third-quarter earnings of $36.1 million ($1.13 a share) on revenues of $1.11 billion compared with earnings a year earlier of $62.4 million ($2.07) on revenues of $1.05 billion. Without a $49.4 million extraordinary gain, third-quarter earnings in 1980 were $13 million (35 cents).

Nine-month earnings were $67.7 million ($2) on revenues of $3.11 billion compared with a net loss of $14.2 million on revenues of $2.85 billion in the same 1980 period. Without the extraordinary gain last year, the loss was $63.7 million ($2.53).

Although third-quarter profits at Lockheed Corp. rose to $20.6 million ($1.18 a share) from $16.4 million ($1.03) last year, Chairman Roy A. Anderson said he is concerned about the depressed airliner market, especially Lockheed's TriStar program.

Sales for the quarter rose to $1.4 billion from $1.3 billion.

Nine-month earnings were $71.7 million ($4.26) in contrast with $1.8 million a year earlier, which amounted to a loss on common stock. This loss reflected a write-off of excess production costs on the TriStar program. Sales were $4.4 billion, up from $3.9 billion.

Gulf & Western Industries Inc. reported yesterday that profits in its 1981 fiscal year ended July 31 rose by 14 percent to $290.9 million ($3.75 a share) from $255.3 million ($3.47) the previous year. Sales were up 7 percent to $5.7 billion from fiscal 1980 sales of $5.34 billion.

Gulf & Western said earnings for 1981 included gains of $34.6 million from sale of investments in securities and losses of $49.9 million from closing or selling unprofitable operations.

Fourth-quarter earnings were $68 million (87 cents) on sales of $1.52 billion compared with fiscal 1980 fourth-quarter earnings of $63.6 million (87 cents) on sales of $1.32.