This is a legitimate sentence but one that could benefit from a little polishing.
We will plant saplings in our school and nearby area, followed by a[n]
exhibition, poster competition, and a speech by our MLA on "Sav[ing]
A native speaker would not construe it to mean that you will plant an exhibition after you plant the saplings. A native speaker would understand that these are distinct activities, despite the fact that your first clause does not put a name on the activity but expresses it with a verb phrase, "we will plant". However, it is best when we don't rely upon the good graces of the reader or listener, expecting them to make little mental adjustments to our words.
Parallelism is a practical way of organizing your sentence to make its meaning as clear as can be. What you want is four nouns, four names for activities. You would want the first clause to contain a nominal, a name for something, just like "exhibition", "competition", and "speech".
There will be a symbolic planting of saplings, followed by an
exhibition, a poster competition, and a speech...
or you could simply bring in the word "and" to link two independent clauses:
We will plant saplings, and then there will be an exhibition, followed by a poster competition, and finally a speech...