An apparent octave (or fifth) occurs when a note is double but is not intended to create another "voice." For example, in strict four-part harmony, one tries to keep all four voices independent so the listener is able to hear four moving melody lines. The "bad" parallel octaves (or fifths) occur when two voices undergo parallel movement in octaves (or fifths.) For example, moving from a C major chord to a D major chord could be problematic. C-E-G-C followed by D-F#-A-D doesn't give the impression of 4 voices, rather it sounds like a single (somewhat complex) voice.
In the example quoted, there is a melody voice with a piano accompaniment to be filled in. In such a case, the piano may play sections using a different number of voices. Often anywhere from 2 to 5 voices are common in such accompaniments. One has to be careful that there is a consistent 3 or more voice texture but some voices may be doubled. One rarely gets away with doubling the bass or the melody but in a full orchestra, lots of doubling happens.