parted, fdisk, lsblk and other commands will give you the total space used by partitions, and the total space on the disk. You will need to calculate what is available for new partitions.
To look into each partition, you will have to mount them.
But gparted, for example, will give you all what you need, but it’s in graphic. gparted does not mount the partitions for what I can see on my pc.
[Edit]
It seems parted can do part of the job 
sudo parted /dev/sdc print free
Model: ATA ST2000VN004-2E41 (scsi)
Disk /dev/sdc: 2000GB
Sector size (logical/physical): 512B/4096B
Partition Table: gpt
Disk Flags:
Number Start End Size File system Name Flags
17,4kB 1049kB 1031kB Free Space
1 1049kB 315GB 315GB ext4 D
2 315GB 629GB 315GB ext4 P
3 629GB 944GB 315GB ext4 P2
4 944GB 1258GB 315GB ext4 M
1258GB 2000GB 742GB Free Space
df -h
can give you free space inside partitions, but only for the mounted ones.
df -h
Filesystem Size Used Avail Use% Mounted on
dev 3,4G 0 3,4G 0% /dev
run 3,4G 1,4M 3,4G 1% /run
efivarfs 128K 110K 14K 90% /sys/firmware/efi/efivars
/dev/sda2 20G 13G 5,4G 71% /
tmpfs 3,4G 0 3,4G 0% /dev/shm
/dev/sda1 50M 3,5M 46M 8% /boot/efi
/dev/sda3 91G 37G 49G 43% /home
tmpfs 3,4G 8,0K 3,4G 1% /tmp
tmpfs 689M 100K 689M 1% /run/user/1000
Same thing fo lsblk -f
lsblk -f /dev/sdc
NAME FSTYPE FSVER LABEL UUID FSAVAIL FSUSE% MOUNTPOINTS
sdc
├─sdc1 ext4 1.0 D 0a9915cd-xxxx 115,1G 55% /media/D
├─sdc2 ext4 1.0 P b7ad1997-xxxxxx 193,4G 28% /media/P
├─sdc3 ext4 1.0 P2 ab3286cc-xxxx (unmounted partition)
└─sdc4 ext4 1.0 M 0b6aabf9-x 239,1G 12% /media/M