When Alberto Abed was 14 years old his father died. The teen-ager dropped out of school and learned to fly planes to help support his family.

Though he was under age, he persuaded instructors at a flying school at Benito Juarez International Airport in Mexico City to give him lessons; he paid for the instruction by cleaning planes, putting air in tires and helping with engine repairs.

His passion for flying has never faded. At a time when few new carriers have emerged worldwide and several in the United States have gone bankrupt, Mr. Abed, who is now 40, has built an airline called Taesa into Mexico's third largest, after Aeromexico and Mexicana Airlines. Some analysts who watch the Mexican market predict that Taesa could eventually move into the No. 2 spot. Low-Cost Flier

Lower costs are what distinguish Taesa from its larger rivals. It has been able to undercut the competition by buying or leasing planes cheaply because of a glut, by keeping its management team small and by hiring nonunion workers for lower pay than its unionized rivals can.

With the financial help of Carlos Hank Rhon, a powerful Mexican industrialist, Taesa has grown from a handful of employees and one plane four years ago to 2,000 employees with a route network that stretches into the United States, Canada, Europe and the rest of Latin America, served by 80 planes. Mr. Abed hopes the carrier will begin to break even by the end of this year.

Taesa expects to carry two million passengers this year -- as many as a carrier like Sabena Belgian World Airlines. That will give it about 9 percent of the overall market of air traffic to, from or within Mexico, as competitors like Aeromexico pull in their horns.

In fact, some airline specialists believe Taesa crowds the field.

"In five years, there will probably not be three major Mexican carriers," said Robert Booth, an aviation consultant who specializes in Latin America at Aviation Management Services in Miami.

Mexicana's high costs make it the most vulnerable, Mr. Booth said, though it has begun to consolidate some operations with Aeromexico.

He added that Taesa could capitalize on a boom in travel that could come from the North American Free Trade Agreement between the United States, Mexico and Canada.

So far, Taesa, which stands for Transportes Aereos Ejecutivos S.A., has mainly been a charter airline, with flights linking American and Canadian cities with Mexican destinations like Cancun, Mexico City, Acapulco and Puerto Vallarta. Of its 90 to 100 flights a day, 43 percent are scheduled and 43 percent are chartered; the rest is cargo service.

But it is trying to expand its scheduled service, without falling into the traps that have hurt many startups. To become major carriers, start-ups often load up with too much debt or are trounced as the big carriers flood routes with flights and cut fares. Consolidation and Smoothing

At a Manhattan restaurant recently, Mr. Abed, who continues to serve as a pilot for the airline he runs, left the table with an aide to take a call from the Governor of Tabasco State in Mexico. Back at the table, he lifted his soda for a toast; Taesa had just gotten permission to fly another route in Mexico, giving the airline more than 30 destinations there.

Still, Mr. Abed said, at the end of this year Taesa will slow its growth to consolidate its new routes and smooth operations.

Taesa's pell-mell expansion has already come at a price. Last spring, the Canadian Government revoked the carrier's right to fly to Canada for a short period while it investigated safety violations.

The Federal Aviation Administration also sent inspectors to locations throughout Taesa's system. Some violations were found, like too much weight on some flights, but the agency backed off when the carrier scrambled to correct the problems.

Paul Kleespies, the president of Asti tours, which specializes in vacation packages to Mexico, said he uses Taesa's Boeing 757's for about 70 percent of his flights mainly because of the carrier's attractive prices. Taesa's six-day trip to Cancun, including six nights in a three-star hotel, costs $329. A similar tour on one of Aeromexico's regularly scheduled flights costs $349, Mr. Kleespies said.

As a result, Taesa will carry about 180,000 charter passengers to Mexico this year from Kennedy International; Newark International; MacArthur Airport in Islip, L.I., and Stewart Airport in Newburgh, N.Y.

Taesa's scheduled flights are competitive in price with the bigger Mexican airlines, though Taesa usually flies to locations not served by its rivals. Its United States destinations from Mexico City include Laredo, Tex.; Las Vegas, Nev., Vail, Colo., and Chicago.

With the deregulation of the Mexican airline industry in 1988, Mr. Rhon, the industrialist, tried to take over Mexicana Airlines. But his bid failed, and he put Mr. Abed in charge of building a new carrier, using Air Taxi, a charter-flight service Mr. Abed ran with his borther, as its base.

They pooled about $500,000 to begin Taesa, with Mr. Abed putting up $150,000 for a 49 percent stake and Mr. Rhon, whose Hermes holding company has about $1 billion in sales, putting up the rest.

Air Taxi's business boomed when, to attract new business, Mr. Abed cut prices for business fliers in half, to about $900 an hour.

At the same time, large carriers were taking delivery of new planes and phasing out less efficient aircraft, creating a glut of used planes.

Using the money made from Air Taxi service, which by 1990 had grown to 25 airplanes, Taesa began to operate charters, thinking that scheduled service would be the next step.

"We saw a niche for ourselves," Mr. Abed recalled.

He and his top executives made the circuit of tour operators and trade shows and bought eight 727-100's for $3.5 million each to run charter flights. His strategy was simple: "If it's a good opportunity, we get the airplanes."

As the surplus grew, large leasing companies like the GPA Group of Ireland began to offer very low rates. Taesa made a deal to lease two new Boeing 757's for $350,000 a month each and eight 737's at $200,000 each.

Later, Taesa leased its first jumbo jet, a Boeing 767, which it uses for charters from Mexico City to seven cities in Europe. Concern Over Size

What worries Mr. Abed now is how to keep his management and operations from getting too bloated as Taesa grows.

That is partly why he serves as a pilot on the weekends. He sees first hand how well and efficiently things are done.

"It gives me a chance to keep in touch with the pilots and to see what's going on in the company," he said. "We want to be keep tight and be efficient."

Photo: Alberto Abed in the cockpit of a Taesa Boeing 757 last month. Mr. Abed, 40 years old, has built Taesa into Mexico's third-largest airline company. Some analysts project that Taesa could edge up to second place. (Sergio Dorantes for The New York Times) Graph: "How Mexican Airlines Compare" shows estimated share of air traffic to, from or within, Mexico: 52% -- Non-Mexican carriers 21.6% -- Mexicana (50% domestic; 50% international) 17.8% -- Aeromexico (60% domestic; 40% international) 8.6% -- Taesa (80% domestic; 20% international) Based on revenue ton-miles (RTM) of both passengers and cargo, scheduled flights, year to October 1992. (Source: Aviation Management Services)