St. Thomas' second proof for the existence of God is an argument from the nature of the efficient cause. He begins with the fact that there exists in the world an order of efficient causes. He goes on to say that we know of nothing that is the efficient cause if itself. Indeed, this would be impossible because for something to be the efficient cause of itself it would need to exist before it could cause itself to exist, which is absurd. Now, it is impossible for us to go to infinity in efficient causes, since if there was no first efficient cause there would be no subsequent efficient causes and thus there would be no effect, which is plainly false. Thus there must be some uncaused cause, which is God.
A problem often arises in how people think about the second way. Since the proof depends on the fact that there cannot be an infinite chain of efficient causes, people often think that it is speaking of God as the first link in the causal chain of the universe. God started the big bang and everything went on from there. This is not the proper way to think about the second way.
As readings in the rest what has come to be called St. Thomas' treatise on God will show, St. Thomas is primarily thinking of God as the efficient cause of existence. Thus God cannot be viewed as simply the first link in the causal chain of the universe. Rather, God is the cause of existence for any and every link in any and every causal chain. He is the cause of the existence of the agent, the action and the effect. Thus God does not cause you simply because He is the cause of the big bang, without which you would not exist. God is the immediate efficient cause of your act of existence.
Nothing in this is opposed to the efficient causality of your parents in your coming to be. The efficient causality of your parents and the efficient causality of God are not incompatible. The causality of your parents and the causality of God are causes on two different levels of reality, physical and metaphysical.
The important thing to remember is that the entire causal chain on the physical level of reality is dependent on the metaphysical causality of God. God is the cause of the existence of your parents. He is the cause of the existence of the conjugal act in which you were conceived. He is the cause of your existence. God is the immediate cause of the existence of everything that exists in the physical world, both substance and accident.