Yes, paraformaldehyde can be used to fix cells. In normal circumstances, formaldehyde is a cross-linking fixative that is used to fix protein and DNA.
Looking at formaldehyde fixation, the following are applicable:
Formaldehyde cross-links with the amino group of the N-terminal amino acid residue and the side chains of 12:
Arginine
Cysteine
Histidine
Lysine
Depending on the peptide sequence, methylol groups, Schiff-bases, and methylene bridges were form
Please note that although the term paraformaldehyde is used, paraformaldehyde exists only as a solid. Once the bonds of the polymer are broken, solutions contain only formaldehyde. Any paraformaldehyde in the solution would simply be an undissolved precipitate at the bottom of the solution.