"As a(n) X", followed by a statement not from the perspective of X

The first sentence in your first example makes no mention of any parents, so it is not clear that it was they, the parents, who were on tour. There is a potential problem in your second example, but it’s imagined rather than real. This syntactical feature is more usually identified as the ‘dangling participle’ problem, where the ambiguity of the reference of the present or past participle of the verb can have strange consequences, as in this example:

Now damaged in the stern, the captain ordered the ship back to port.

Such instances are good for a joke, but, in practice, the context usually makes the meaning perfectly clear.