  We present a family of novel Lax operators corresponding to representations
of the RTT-realisation of the Yangian associated with $D$-type Lie algebras.
These Lax operators are of oscillator type, i.e. one space of the operators is
infinite-dimensional while the other is in the first fundamental representation
of $\mathfrak{so}(2r)$. We use the isomorphism between the first fundamental
representation of $D_3$ and the $\mathbf{6}$ of $A_3$, for which the degenerate
oscillator type Lax matrices are known, to derive the Lax operators for $r=3$.
The results are used to generalise the Lax matrices to arbitrary rank for
representations corresponding to the extremal nodes of the simply laced Dynkin
diagram of $D_r$. The multiplicity of independent solutions at each extremal
node is given by the dimension of the fundamental representation. We further
derive certain factorisation formulas among these solutions and build transfer
matrices with oscillators in the auxiliary space from the introduced degenerate
Lax matrices. Finally, we provide some evidence that the constructed transfer
matrices are Baxter Q-operators for $\mathfrak{so}(2r)$ spin chains by
verifying certain QQ-relations for $D_4$ at low lengths.
