The parquet you have is too thin to refinish. All the cracking is from groove side of the flooring. It will continue to crack as you use the floor. It needs to be replaced.

It's hard to tell, but it looks like cheap Rubberwood, marketed as Hevea. These were most often sold as 5/16" thick squares with tongues and grooves. Your crackling is either coming from dried up adhesive, or if it's a foam-backed parquet, the application of urethane will cause crackling as the boards rub against each other when walked on. The latter will somewhat subside with time. Maybe you can live with it for a while, but ultimately it would require replacement,