Understanding Linear Functions: Identifying Parallel Lines and Their Representations

To solve this problem, we'll examine each given choice:

• Choice 1: $y = -x$ and $y = x$. Both can be written in the form $y = mx + b$. Slopes are $-1$ and $1$, hence not parallel.

• Choice 2: $y = 1 + x^2$ and $y = 2 + x^2$. These are quadratic forms, not linear equations.

• Choice 3: $y = 2(x+1)-x$ simplifies to $y = (2-x) + 2$, which further reduces to $y = x + 2$, hence $y = x + 2$.
  - Both equations, $y = x + 2$ and $y = x + 2$, are in linear form with equal slopes of $1$. They are the same line, hence parallel by default.

• Choice 4: $y = 2 + x$ is the same as $y = x + 2$. - $y = x$ compares with $y = x + 0$.
  - Slopes of both are $1$, hence they are parallel.

• Choice 5: Claims C and D are correct, which entails verifying that both choices depict linear functions and parallel lines as previously identified.

Upon analysis, choices C and D both represent linear functions and their line pairs have equal slopes, indicating parallel lines. Thus, the correct answer is that both choices C and D are correct.

Therefore, the correct answer to the problem is: Choices C and D are correct.