No. Another manufactured myth.
Oxfordians like to claim that this is a monument to a grain or wool dealer, or to Shakespeare's father, John. The claim is based on a picture which was originally a rough sketch of Shakespeare's monument made by William Dugdale intended for publication in his Antiquaries of Warwickshire 1656. He wasn't any kind of an artist and so passed his sketches on for improvement to an artist called Wenceslas Hollar.
Hollar's work was then copied by Gerard Van der Gucht for an engraving for Nicholas Rowe's account of Shakespeare's life of Shakespeare in 1709. Arguments that is is an image of a grain or wool dealer are simply mistaken. The picture is an engraving of a drawing of a rough sketch which is the reason why it looks so different from the bust we see today.
The pen is missing from the sketch but has gone missing many times during the centuries and may not have been there when Dugdale visited the church. In any case, Shakespeare's father was almost certainly a Catholic recusant and was fined a couple of times for non-attendance at church, making it inconceivable that he would be honoured in this way.
The argument that it was changed before the much more accurate illustration commissioned by Alexander Pope in 1723 takes no account of the fact that there has always been an inscription at the base describing the figure as a great scholar and writer.
While we're on the subject of Shakespeare's father being fined for non- attendance at Church, the names of three others fined at the same session of the Religious Commissioners are significant: Bardolph, Fluellen and Court. These are all characters in Henry V. So another little detail that connects Shakespeare with Stratford and an interesting insight into how writers collect names for their characters.
If you visit Holy Trinity and stand below the actual monument, the reasons for the shortcomings of Dugdale's drawing become even clearer. The monument is high up so that from below, the perspective is extreme and the bust looks not unlike the drawing. Dugdale has corrected the straight lines and used right angles on the carved niche where he knows them to be.
The unusual features of Dugdale's sketch can all be accounted for by his lack of knowledge of 3-point perspective.