To prove by contradiction that "In the same plane, if $a\bot c$, $b\bot c$, and $a$ is parallel to $b$", we start by understanding the structure of a proof by contradiction. This method involves assuming the opposite of what we want to prove and then showing that this assumption leads to a contradiction.

Step 1: Identify the statement to be proved, which is "if $a\bot c$, $b\bot c$, and $a$ is parallel to $b$".

Step 2: To use proof by contradiction, we assume the opposite of the conclusion we are trying to reach. In this case, the conclusion that would naturally follow from the premises is that $a$ and $b$ are parallel (since $a\bot c$ and $b\bot c$ suggest $a$ and $b$ share a common relationship to $c$, and being parallel to each other is consistent with this).

Step 3: The opposite of the conclusion (that $a$ and $b$ are parallel) is that $a$ and $b$ are not parallel. This is what we assume for the sake of contradiction.

Step 4: By assuming $a$ and $b$ are not parallel, we are directly contradicting the given conditions combined with the properties of geometric figures in a plane. This contradiction indicates that our assumption must be false.

Step 5: Since our assumption (the opposite of the conclusion) leads to a contradiction, the original conclusion (that $a$ and $b$ are parallel if $a\bot c$ and $b\bot c$) must be true.

Therefore, the correct answer, following the structure of a proof by contradiction, is to assume that $a$ and $b$ are not parallel.

Final Answer: $\boxed{A}$.