This is a very important question to understand. We usually don't see two consecutive unisons, fifths, or octaves followed by one another (with some exceptions) in classical tonal music (not so in pop music and jazz - parallel fifths are quite common).
To answer this, we have to look deeper into the theory of the art of counterpoint in the Renaissance. Specifically, let's see what Gioseffo Zarlino, the great 16th century Italian music theorist and composer had to say about it in his 1558 treatise the Art of Counterpoint:
"They [ancient composers] realized that harmony results from things that are diverse, discordant, and contrary to each other rather than alike in every way. If harmony is born from this variety, it follows that in music not only must the parts be distant from one another with respect to pitch, but their movements should be different, and they should contain diverse consonances of various ratios. The more harmonious a composition seems to us, the more variety we will discover in it: in the vertical distances between its parts, in its movements, and its proportions."
So what Zarlino is really saying is that parallel intervals inhibit the independence of voices. By the way, parallel fifths were perfectly acceptable in the late Middle Ages - the age of organum by such composers of the School of Notre Dame as Leonin and Perotin (12th-13th centuries).
Around that time organs were tuned in pure perfect fifths (the Pythagorean temperament) and the thirds sounded out of tune and were avoided. If you want to imagine this style, just take any tune of Gregorian chant and supply the second voice in parallel fifths above the melody and you will start to hear some old and archaic sounds - something similar to the 12th-13th century music (though it was more complex than simple parallel fifths at that time).
What about the parallel thirds in the art of counterpoint? Well, even the thirds cannot be both major (F-A, G-B) because the ear perceives the dissonant augmented fourth F-B between the outer voices in these two intervals very clearly. The thirds were best used as major and minor in alternation.
Ach Gott vom Himmel sieh darein by Jan Pieterszoon Sweelinck (1562-1621), the Great Orpheus of Amsterdam, also called Maker of German Organists (Deutsche Organistenmacher).
Jesus Christ, Our Blessed Savior