We consider codimension two spacelike submanifolds with a parallel normal field (i.e. vanishing normal curvature) in Minkowski space. We use the analysis of their contacts with hyperplane and hyperquadrics in order to get some global informations on them. As a consequence we obtain new versions of Carat¥`eodory's and Loewner's conjectures on spacelike surfaces in $4$-dimensional Minkowski space and $4$-flattenings therorems for closed spacelike curves in $3$-dimensional Minkowski space.