Difference in Differences - why do we use the terms 'control group' and 'treatment group'? When doing Difference in Differences, we basically pretend to know the average treated outcome $\frac{\sum_{i=1}^n Y_i(1)}{n}$ and the average no-treatment outcome $\frac{\sum_{i=1}^n Y_i(0)}{n}$ of an initial group of units (by assuming a parallel counterfactual trend with a secondary group of units). Therefore, we can directly use the fact that the sample ATE is an unbiased estimator of the true ATE to estimate the true ATE by $\frac{\sum_{i=1}^n Y_i(1)-Y_i(0)}{n} = \frac{\sum_{i=1}^n Y_i(1)}{n} - \frac{\sum_{i=1}^n Y_i(0)}{n}$.
This is how I rationalize the Difference in Differences result. The parallel trend assumption spares us the effort of dividing the units into treatment and control groups and it means that there is no selection bias to worry about (which Wikipedia confirms)
LE: To clarify my reasoning, I painted this graph.

By making the Parallel Trends assumption and observing that the second group reaches point a, we automatically know that the first group would have reached point c had it not been for the treatment. We thus have everything we need to know about the first group:

*

*the average (observed) treatment outcome = $\frac{\sum_{i=1}^n Y_i(1)}{n}$ = d

*the average (assumed) no-treatment outcome = $\frac{\sum_{i=1}^n Y_i(0)}{n}$ = c

Therefore, the sample ATE is d-c, which is an unbiased estimate of the true ATE. There are no treatment or control groups, because we don't make any assignment. We literally know or assume both average potential outcomes of the first group.
 A: It seems like your question is asking why we use two-group, two-time point designs rather than one-group, two-time point designs. Why would we need a control group when we have the outcomes both under no treatment and under treatment for the group that receives treatment?
The answer is that the outcomes in the pre-period are not good estimates of the potential outcomes of the post-period had the units not been treated. If there is a natural trend for the outcomes in the absence of treatment (e.g. a maturation effect), only having one group would cause one to mistake the natural trend for a treatment effect. Including a control group that follows that natural trend allows one to subtract off the natural trend observed in the control group from the trend observed in the treated group and purify the portion of the trend due only to treatment. This relies on the assumption that the treated units would have followed the same trajectory as the control units had they not been treated (this is the parallel trends assumption). Without the control group, you have no way to defend your estimate from the argument that the treated units would have followed the observed trajectory even in the absence of treatment.
For example, let's say I'm testing the efficacy of a new drug against the common cold. I take a bunch of people with the cold, measures their symptoms on some continuous scale, give them my drug, and measure their symptoms a week later. I find that all my patients recovered; their cold symptoms are now close to zero. Was my drug effective? We don't know what would have happened had they not received the drug. Colds tend to dissipate in about a week anyway, so how do we know the drug caused the change? Only including a control group and measuring their symptoms along with the treated units would allow me to answer that question. Had I included a control group and found that their outcomes followed an identical trajectory to those of the treated group, I would not be able to claim my treatment was effective; the observed change in the treated group from pre to post was due simply to maturation.
A: I figured out the reason of my confusion.
We can interpret the Diff-in-Diff technique in two ways:

*

*Only consider the members of the first group to be units in the experiment.
This was my initial interpretation. In this case, we don't split the units in control and treatment sub-groups because we don't need to. We have both average possible outcomes under the Parallel Trends assumption, so we can directly estimate the ATE through the sample ATE (which is unbiased), as I described in the question. Note that here we didn't do any selection.

*Consider the members of the first group and the second group to be units in the experiment. In this case, the first group is the treatment group and the second group is the control group. And we know that the difference between their observed average outcomes is also an unbiased estimator of the ATE, but only if the selection bias is 0. Which it is. (The selection bias is defined as the difference in the expected outcome without treatment of the treatment group and the control group. The Parallel Trends assumption essentially tells us that this difference is 0.) Note that here we did a selection, but its bias is 0.
Both interpretations reach the same ATE estimate, but for different reasons.
