Is reed switch sensitive to DC magnetic field?
How would I know if a reed switch in parallel to a DC wire in which is flowing current can be closed by the magnetic field produced?


You need to create a reed relay to create a strong enough magnetic field along the reeds. Running a wire alongside the reed switch won't work.



Source: How to Choose the Right Relay by National Instruments.

A minimum number of ampere-turns will be required and this must be calculated from the manufacturer's datasheet. Hamlin's Application Note AN102A may be of interest along with some other links below.




  ... what's the minimum current required?


It's the product of the current and the number of turns. More turns means less current is required.

Links

Reed Sensors vs. Hall Effect Sensors by Medor Electronic.

Basic Electrical Parameters of Reed Switch
Products by Meder Electronic.